Magnon-Phonon Coupling in Ferromagnetic Thin Films

Transcription

1 TECHNISCHE UNIVERSITÄT MÜNCHEN WALTHER- MEISSNER- INSTITUT FÜR TIEF- TEMPERATURFORSCHUNG BAYERISCHE AKADEMIE DER WISSENSCHAFTEN Magnon-Phonon Coupling in Ferromagnetic Thin Films Diploma thesis Matthias Pernpeintner Advisor: PD Dr. Sebastian T. B. Gönnenwein Garching, October 2012

2

3 Contents Introduction 1 1 Surface acoustic waves on piezoelectric substrates Equations of motion Boundary conditions Surface-confined plane-wave solutions Surface acoustic waves on YZ-LiNbO 3 and 36 YX-LiTaO Material constants Rayleigh Waves on YZ-LiNbO Shear horizontal waves and Rayleigh Waves on 36 YX-LiTaO Summary Modelling of acoustically driven FMR Landau-Lifshitz-Gilbert approach Landau-Lifshitz-Gilbert equation and free-enthalpy density Magnetization precession and virtual driving field Absorbed radio-frequency power Effects of shear strains on ADFMR symmetry Backaction of ADFMR on the SAW Surface acoustic wave model Equations of motion and plane-wave ansatz Interface and boundary conditions Calculation results Summary Acoustically driven FMR: Experimental study and comparison to theory Samples Experimental setup Data processing I

4 II Contents SAW transmission SAW power Normalized transmission coefficient and absorbed power Rayleigh wave driven ADFMR SAW transmission of Sample Comparison of the experimental data with the ADFMR models Angle-dependent ADFMR Comparison of shear-horizontal and Rayleigh wave ADFMR SAW transmission of Sample Angle-dependent ADFMR Comparison of the experimental data to the SAW Model Summary Tuneable surface acoustic wave resonator SAW resonators and overview Flipchip delayline Tuneable delayline Tuneable resonator Transmission overview Application of a time gate SAW transmission and comparison to conventional resonator SAW transmission as a function of V piezo Summary Summary and outlook Summary Outlook A Determination ofthe superposition coefficients A α and B α in the surface acoustic wave model 99 B Effective reduction of the magnetoelastic coupling constant b 1 by surface pinning of electron spins 101 C Samples 107 Bibliography 111

5 Introduction Since the famous experiment by Stern and Gerlach [1,2] was performed for the first time in 1922, the spin of electrons and nuclei has been subject to extensive experimental and theoretical investigation. The proof of the spin-statistics theorem by Pauli in 1940 [3] and the discovery and determination of the anomalous magnetic moment of the electron [4,5] in the 1950s are only two examples of important contributions to the understanding of spin and spin-related phenomena, which also have had an enormous impact on physics in general. Besides, spin-related physics has become more and more relevant in application and industry: Nuclear Magnetic Resonance (NMR), for example, first realized in 1946 by Bloch [6,7] and Purcell [8], is nowadays an important tool for the imaging of organs and tissue in medical diagnosis as well as for the non-destructive material analysis in applied science. The giant magnetoresistance (GMR) effect, discovered by Grünberg [9] and Fert [10] in 1988, is exploited in today s hard disks to read out large amounts of data efficiently. In recent years, the fields of spin electronics (short: spintronics), spin caloritronics and spin mechanics have been established, which deal with the systematic control of electron/nucleon spins via electric/magnetic fields, temperature gradients and elastic properties of solids [11 14]. A detailed understanding of the coupling of spins to their environment and the development of distinct controlling mechanisms could be the basis for a fundamental change in information technology, where data processing, transfer and storage are based on electron spins instead of charges [15]. This would enable further miniaturization of computing devices involving increasing computing power and storing capacities as well as a reduction in energy consumption and heat generation. The present thesis deals with a particular issue of spin mechanics, namely the coupling between quantized lattice vibrations, i. e. phonons, and coherent spin excitations in a ferromagnet, called magnons. This magnon-phonon coupling was studied using hybrid devices which consist of a piezoelectric crystal onto which a ferromagnetic thin 1

6 film was deposited. On the surface of the piezoelectric crystal, surface acoustic waves (SAWs) are excited, which induce radio-frequency strains in the ferromagnetic film. These strains interact with the magnetization in the FM via magnetoelastic coupling and excite a so-called acoustically driven ferromagnetic resonance (ADFMR) [16,17]. In this thesis, the coupling between SAW and magnetization was studied experimentally and theoretically. Chapter 1 gives an analytical description of the SAW propagation on piezoelectric crystals. It is shown that the type of surface wave is determined by the material properties as well as by the surface cut direction of the used crystal. Moreover, we demonstrate that it is possible to excite different SAW types on one and the same crystal, which can be distinguished by their propagation speed. Chapter 2 presents three modelling approaches to ADFMR, based on an effective field ansatz which considers magnetoelastic coupling via its contribution to the freeenthalpy density of the FM. These models differ in their range of validity as well as in their numerical complexity. Besides, it is shown that ADFMR qualitatively differs from conventional FMR [18, 19]. Chapter 3 deals with the experimental investigation of ADFMR. We measure the change in complex transmission due to ADFMR as a function of orientation and strength of the external static magnetic field for different frequencies. We use different surface waves types to excite ADFMR and compare their characteristic signatures. It is shown that the presented ADFMR models are consistent with one another and able to quantitatively describe the experimental results. Besides the experimental and theoretical investigation of ADFMR, preliminary work for future ADFMR experiments with SAW resonators has been done. Resonators for surface acoustic waves are well-known and have been used, e. g., as narrowband frequency filters for about 40 years [20 24]. Yet, we show that for our purposes it is necessary to have a possibility to in-situ tune the resonator, which means to change its acoustic length by an external control parameter. Thus, the so-called flipchip design was chosen, which allows to mechanically shift the reflectors of the resonator against each other. In Chapter 4, the working principle and the fabrication of the tuneable resonator and preliminary experiments are shown. Furthermore, the tuneable resonator is characterized and its functionality is demonstrated. In Chapter 5, the obtained results are summarized and several ideas for further 2

7 studies of ADFMR and related phenomena are presented together with some preliminary calculations. Finally, we would like to mention that the SAW experiments, together with the theoretical modelling performed in collaboration with L. Dreher and M. Brandt at the Walter-Schottky-Institut of the Technical University Munich have resulted in a manuscript submitted to Physical Review B [17]. 3

8 4

9 Chapter 1 Surface acoustic waves on piezoelectric substrates In this thesis, acoustically driven ferromagnetic resonance (ADFMR) [16] is investigated using different types of surface acoustic waves (SAWs) propagating on substrate/ferromagnetic thin film bilayers. Depending on the type of acoustic wave and the corresponding strains induced in the FM thin film, qualitatively different ADFMR signatures arise as will be shown in Chap. 3. Here, we first calculate the wave types propagating on the used piezoelectric crystals and determine the strains induced. The derivation hereby follows [25] and [23]. 1.1 Equations of motion We start with the most general form of the equations of motion for elastic deformation in a solid without external forces [25, 26]: ρ 2 u i t 2 = j σ ij x j, i,j {1,2,3}, (1.1) with density ρ, elastic deformation u i and stress tensor σ ij. In a piezoelectric crystal, σ ij isafunction ofthe stiffness tensor c ijkl, thestraincomponents ε kl, the piezoelectric tensor e ijk and the electric field E: σ ij = k,l c ijkl ε kl k e kij E k (1.2) 5

10 6 Chapter 1 Surface acoustic waves on piezoelectric substrates with ε kl = 1 2 ( uk + u ) l. (1.3) x l x k The electric displacement D in a piezoelectric material is given by D i = j ǫ ij E j + j,k e ijk ε jk, (1.4) where ǫ ij denotes the dielectric tensor. As the velocity of elastic deformation propagation (i.e. the speed of sound) is about five orders of magnitude smaller than the speed of light, we can assume the electric field to be quasi-static so that it can be written as the gradient of a potential function φ: E = φ. (1.5) Using Eqs. (1.2) and (1.5), the equation of motion (1.1) becomes ρ 2 u i t 2 = j,k [ e kij 2 φ x j x k + l c ijkl 2 u k x j x l ]. (1.6) As we are dealing with insulating substrates, there are no free charges and Maxwell s equations give us divd = 0. Applying this to Eq. (1.4) leads to [ 2 φ ǫ ij x i,j i x j k e ijk 2 u j x i x k ] = 0. (1.7) Equations (1.6) and (1.7) yield four equations relating the four unknown quantities u and φ. Together with appropriate boundary conditions, this allows a complete description of the coupled acoustic and electrical wave in a piezoelectric solid. 1.2 Boundary conditions In the following, the piezoelectric crystal is modelled as an anisotropic, piezoelectric medium with infinite extent in the x- and y-directions and a surface at z = 0. The half-space z 0 is filled by the crystal, while z > 0 is vacuum. In this model, appropriate boundary conditions have to be specified for the surface at z = 0. First, we have the mechanical boundary condition for a free surface, that

11 Section 1.3 Surface-confined plane-wave solutions 7 accounts for the fact that there are no stresses on the surface of the solid: σ xz = σ yz = σ zz = 0 at z = 0. (1.8) Second, for piezoelectric materials, an electrical boundary condition is necessary which defines the electrical field at the surface. Usually two qualitatively different cases are considered: First, a free surface, where the space above the substrate is assumed to be vacuum. As there are no free charges at the surface, the normal component of the electric displacement D z has to be continuous. For z 0, D z is given by (1.4), while in the upper half-space (i.e. z > 0), ǫ = 1 and therefore D z = ǫ 0 φ z for z > 0. (1.9) The second case considered is the metallized-surface one: Here, we assume the piezoelectric half-space to be covered with an infinitely thin, perfectly conducting metal. In this case, the electrical potential is constant at the surface z = 0 and per definition set to zero: φ = 0 at z = 0. (1.10) 1.3 Surface-confined plane-wave solutions As we are particularly interested in surface wave solutions of Eq. (1.6) and (1.7), we make a plane-wave ansatz, which decays exponentially within the substrate. For convenience we choose a coordinate system such that x is along the SAW propagation direction, z is normal to the film plane and y is in the film plane so that (x,y,z) is a right-handed frame of reference. Then, the mentioned surface-confined plane-wave ansatz can be written as u(x,t) = u 0 exp(iγz)exp(iβx iωt) (1.11) φ(x,t) = φ 0 exp(iγz)exp(iβx iωt). (1.12) Here, ω denotes the angular frequency andβ the wave number of the SAW, which is assumed to be real. γ describes the z-dependence of the acoustic wave and is complex in general. Using (1.11), we can specify the boundary condition (1.9) for the free-surface case.

12 8 Chapter 1 Surface acoustic waves on piezoelectric substrates As the electrical potential φ is continous at the surface z = 0, it has the form φ = φ 0 f(z)exp(iβx iωt), (1.13) where the z-dependence is given by f(z). In addition, φ(z > 0) satisfies Laplace s equation 2 φ = 0 and has to vanish for z = +, which results in f(z) = exp( βz). Inserting this into Eqs. (1.13) and (1.9) gives D z = ǫ 0 βφ at z = 0. (1.14) To find SAW solutions, the plane-wave ansatz (1.11) is substituted into the equations of motion (1.6) and (1.7), which form a homogeneous system of four equations in the four variables u and φ. Setting the determinant of coefficients D 1 to zero gives a relation between γ and β. In general, D 1 is a eighth-degree polynomial in γ, so there are eight complex solutions for γ, depending on the parameter β. As we are only interested in surface-confined wave solutions, we set the additional condition Im γ < 0, which guarantees exponential decay of the acoustic wave for z and excludes half of the found values. The remaining, valid solutions shall be labeled γ α, with an upper greek index α {1,2,3,4}. In order to get a SAW solution which meets the boundary conditions, a superposition ansatz is made: u tot (x,t) = α φ tot (x,t) = α A α u α (x,t) (1.15) A α φ α (x,t). (1.16) The coefficients A α have to be chosen such that (1.8) and (1.14) resp. (1.10) are satisfied. These conditions lead to another determinant D 2 which has to be zero. D 2, however, will be zero only if the correct value for β has been chosen in the plane-wave ansatz (1.11). So, solving D 2 (β) = 0 iteratively finally gives us a SAW solution which satisfies the equations of motion as well as the boundary conditions. Having found the correct β, the SAW velocity is v = ω/β and the coefficients A α can be determined by solving the linear equation system given by the boundary conditions (1.8) and (1.14) resp. (1.10).

13 Section 1.4 Surface acoustic waves on YZ-LiNbO 3 and 36 YX-LiTaO Surface acoustic waves on YZ-LiNbO 3 and 36 YX-LiTaO 3 LiNbO 3 and LiTaO 3 are widely used in industrial SAW-based applications in the field of communication, signal processing and sensing. Both materials are piezoelectric and pyroelectric insulators, exhibit trigonal crystal symmetry and belong to the 3m point group [27,28]. Because of the anisotropy of LiNbO 3 and LiTaO 3, the crystal cut direction defines if and which acoustic wave can propagate on the surface of the crystal. To specify the crystal cut, the crystal coordinate system (X, Y, Z) is introduced, where Z is along the c-axis of the crystal, X is parallel to one of the a-axes and Y is such that (X,Y,Z) is orthogonal and right-handed. Y-cut LiNbO 3 which means that the surfacenormalisparalleltoy isknowntosupportrayleighwaves 1 alongthez-axis with a sound velocity of about 3488m/s (see Fig. 1.1a) [30,31]. The comparatively high coupling factor 2 κ 2 = 4.5% [30] between electrical and acoustic field allows the efficient excitation of surface acoustic waves via electrical fields applied to the crystal. Therefore LiNbO 3 is very well suited as a SAW-carrying crystal for frequency filters, Fourier convolvers, oscillators etc. Apart from the well-known Rayleigh waves, there is a variety of other surface wave types as, e.g., Love waves or Bleustein-Gulyaev waves, as well as pseudo-surface or leaky surface waves [25]. One technologically interesting wave type is the shear horizontal (SH) wave, whose displacement is mainly in the surface plane and perpendicular to the propagation direction of the wave (see Fig. 1.1b). Shear waves are used for biosensing applications (see e. g. [32 34]) as they propagate also at interfaces to liquid media and are sensitive to the mechanical and electrical properties of the adjacent liquid. Shear horizontal waves can be excited, e. g., on ST-quartz [33], certain piezoelectric ceramics [35] and 36 Y-cut LiTaO 3 3, which has been used in the present work. 1 As illustrated in Fig.1.1a, Rayleigh wavesare characterizedby a phase shift of π/2 between normal and longitudinal displacement component (u z resp. u x ) and a vanishing transverse component u y. Rayleigh waves are named after Lord Rayleigh, who first discovered them in 1885 [29]. 2 The electromechanical coupling coefficient κ 2 is a measure of the coupling between electrical and elastic part of the SAW [30]. If the SAW is excited using interdigital transducers (IDTs), as it is the case here, κ 2 determines the efficiency of the SAW excitation because IDTs directly couple to the electrical field of the SAW only. 3 For 36 -rotated Y-cut X-propagating LiTaO 3, as it is commonly used, the relation between the crystal axes (X,Y,Z) and the sample coordinate system (x,y,z) is rather complicated and can be found in [36].

14 10 Chapter 1 Surface acoustic waves on piezoelectric substrates a b z = Y y = X x = Z Figure 1.1: Schematic, highly exaggerated illustration of (a) Rayleigh and (b) shearhorizontal wave, propagating along the x-axis. The color code shows the real, i. e. physically relevant, part of the mechanical displacement, Re(u). The coordinate system shows the orientation of the crystal axes X, Y and Z and their mappingto the sample frame of reference (x,y,z) introduced in Sect Material constants The elastic properties of a solid are mainly characterized by the stiffness tensor c ijkl and the density ρ. The dielectric tensor ǫ ij describes the response of the solid to electromagnetic fields, while the piezoelectric tensor e ij relates both acoustic and electrical fields. To simplify notation and reduce the number of indices, we use Voigt notation [37, 38] for the material tensors. Then, the stiffness tensor for trigonal 3m crystals has the form [31] c 11 c 12 c 13 c c 12 c 11 c 13 c c 13 c 13 c C =. (1.17) c 14 c 14 0 c c 44 c c 14 (c 11 c 12 )/2 The piezoelectric tensor is given by e = e 15 e 22 e 22 e 22 0 e e 31 e 31 e (1.18)

15 Section 1.4 Surface acoustic waves on YZ-LiNbO 3 and 36 YX-LiTaO 3 11 LiNbO 3 LiTaO 3 Density ρ kg/m 3 Elastic constants c N/m 2 c N/m 2 c N/m 2 c N/m 2 c N/m 2 c N/m 2 Piezoelectric constants e C/m 2 e C/m 2 e C/m 2 e C/m 2 Dielectric constants ǫ ǫ Table 1.1: Material constants for LiNbO 3 and LiTaO 3, taken from [27] and [39], respectively. As the piezoelectric tensor e ij is not given explicitly in [27], it has been calculated using the relation e ij = d ik c kj from [26] (The d ik are called piezoelectric strain constants). Finally, the dielectric tensor is ǫ = ǫ ǫ (1.19) 0 0 ǫ 33 In Tab. 1.1, literature values of the material parameters for LiNbO 3 and LiTaO 3 are listed. All material constants are given in the crystal coordinate system (X,Y,Z) and have to be transformed into the sample frame of reference introduced in Sect Due to the simplified Voigt notation, specific transformation matrices have to be used for the tensors c ij and e ij, which can, e.g., be found in [26] Rayleigh Waves on YZ-LiNbO 3 Using an implementation of the algorithm explained in Sect. 1.3 in Mathematica and the literature material constants from Tab. 1.1, displacements and strains of the

16 12 Chapter 1 Surface acoustic waves on piezoelectric substrates displacement (a.u.) a Re(u z ) Re(u x ) Re(u y ) x ( ) electrical potential (a.u.) b Re( ) x ( ) Figure 1.2: x-dependence of (a) displacement and (b) electrical potential of a Rayleigh wave propagating on YZ-LiNbO 3, assuming a metallized surface. λ is the SAW wavelength, and z was taken as zero. Here, the physically relevant real part of u and ϕ is plotted instead of the magnitude in order to show the phase shift between the u i. Rayleigh wave propagating on YZ-LiNbO 3 have been calculated. In Fig. 1.2, the mechanical displacement u and the electrical potential φ are plotted against the SAW propagation direction x, whereas Figs. 1.3 and 1.4 show u and φ as a function of the surface normal z for metallized and free surface boundary conditions, together with literature data from [25]. In x-direction, we see a sine oscillation with a phase shift of π/2 between the longitudinal (u x ) and normal (u z ) displacement. The transversal component of the displacement, u y, is zero within numerical limits. In z-direction, we observe an exponential decay of displacement and potential on the scale of the wavelength λ, and a zero crossing of u x at z λ/5. Regarding Re(u x ) or Im(u x ) instead of its absolute value reveals that u x changes sign at the mentioned point. This means that the motion of a single particle in the solid changes its rotation direction from clockwise (for z > λ/5) to counterclockwise (for z < λ/5), which can also be seen in Fig Comparing literature data and calculation results, as shown in Figs. 1.3 and 1.4, yields full quantitative agreement. Figures 1.3 and 1.4 illustrate the influence of the electrical boundary condition on the SAW, as they show the calculated u and φ for a metallized and free surface, respectively. As one can see, the change in u is comparatively small, whereas φ differs remarkably near the surface. Moreover, the sound velocity of the SAW decreases by

17 Section 1.4 Surface acoustic waves on YZ-LiNbO 3 and 36 YX-LiTaO 3 13 displacement (a.u.) a calculated literature Rayleigh wave YZ-LiNbO 3 u z metallized surface u y u x z ( ) electrical potential (a.u.) b calculated literature z ( ) Figure 1.3: z-dependence of (a) displacement and (b) electrical potential of a Rayleigh wave propagating on YZ-LiNbO 3, assuming a metallized surface. The solid and dashed line show calculation results and literature data, respectively. λ is the SAW wavelength, z = 0 corresponds to the surface of the crystal. displacement (a.u.) a calculation literature Rayleigh wave YZ-LiNbO 3 u z free surface u y u x z ( ) electrical potential (a.u.) b calculation literature z ( ) Figure 1.4: z-dependence of (a) displacement and (b) electrical potential of a Rayleigh wave propagating on YZ-LiNbO 3, assuming a free surface. The solid and dashed line show calculation results and literature data, respectively. λ is the SAW wavelength, z = 0 corresponds to the surface of the crystal.

18 14 Chapter 1 Surface acoustic waves on piezoelectric substrates sound velocity v met Rayleigh wave 3433m/s strains ε xx 1 ε yy 0 ε zz i ε yz 0 ε xz i ε xy 0 Table 1.2: Calculated sound velocity and strains for a Rayleigh wave on YZ-LiNbO 3 (metallized surface). about 2.5%. The difference in the speed of sound v is crucial for the excitation of SAWs with interdigital transducers as κ 2 = 2 v/v [30]. Having calculated displacement and electrical potential of the SAW, we can substitute u tot and φ tot given by Eqs. (1.15) and (1.16) into Eq. (1.3) to determine the strains induced by the SAW. The calculated values are listed in Tab In Fig. 1.5, the three non-vanishing strain components ε xx, ε zz and ε xz are shown by a color code. If the (bulk) LiNbO 3 crystal is covered with a thin ferromagnetic film, these strains can be taken as an approximation of the strains in the FM film driving the magnetization precession via magnetoelastic coupling [16]. Comparing the calculated sound velocity to the literature value v met = 3410m/s, which will be given in Tab. 3.2, shows the expected agreement Shear horizontal waves and Rayleigh Waves on 36 YX-LiTaO 3 According to [40], 36 Y-cut LiTaO 3 does support not only shear horizontal waves, but also Rayleigh waves, with a distinctly lower speed of sound and a much weaker coupling κ 2 between electrical and mechanical wave component. In sensing devices, the effects of this Rayleigh wave mode are usually neglected. Concerning the study of acoustically driven FMR (Chap. 3), however, 36 Y-cut LiTaO 3 allows to investigate FMR driven by two different SAW types in one and the same sample. In Sect , wewillsee thatthevirtualdrivingfieldhoffmrisproportionaltothestraininduced by the acoustic wave, which is qualitatively different for Rayleigh and SH waves. So

19 Section 1.4 Surface acoustic waves on YZ-LiNbO 3 and 36 YX-LiTaO 3 15 a ε b xx ε zz z c ε xz y x Figure 1.5: Three-dimensional plot of the dominant strain components induced by a Rayleigh wave on YZ-LiNbO 3. The propagation direction of the SAW is along the x-axis, z is the surface normal. The color code shows the real part of (a) ε xx, (b) ε zz and (c) ε xz, where red and blue color denotes positive and negative strain, respectively. The scaling of the color code is the same for the three figures. Similar to Fig. 1.1, the displacement has been exaggerated strongly.

20 16 Chapter 1 Surface acoustic waves on piezoelectric substrates displacement (a.u.) a Rayleigh wave 36 YX-LiTaO 3 uz u y u x z ( ) electrical potential (a.u.) b z ( ) Figure 1.6: z-dependence of (a) displacement and (b) electrical potential of a Rayleigh wave propagating on 36 YX-LiTaO 3, assuming a metallized surface. λ is the SAW wavelength, z = 0 corresponds to the surface of the crystal. we expect the signature of ADFMR to depend on the type of surface wave. InFigs. 1.6and 1.7, uandφofbothrayleighandshwave areplottedasafunction of z, assuming a metallized surface of the LiTaO 3 crystal. One can see that for the SH-wave case the u y -component of the SAW dominates, accompanied by a small contribution of longitudinal and normal displacement. In the Rayleigh-wave case, the elastic part of the SAW, i. e. the displacements, looks quite similar to the Rayleigh wave on LiNbO 3, while the electrical potential differs remarkably (see Fig. 1.3). Of course, we do not expect a perfect agreement as the material constants of LiNbO 3 and LiTaO 3 differ considerably (cf. Tab. 1.1). Regarding the calculated strains listed in Tab. 1.3, the shear strain ε xy dominates in the SH wave case, whereas for the Rayleigh wave case ε xy is small compared to the longitudinal strain ε xx. For both wave types, we observe phase shifts close to π/2 or π between the dominant strain components. This can also be seen from Figs. 1.8 and 1.9, where the dominant strain components are plotted in a three-dimensional representation similar to Fig The calculated strains as well as the displacements and the electrical potential shown in Figs. 1.6 and 1.7 demonstrate the differing character of Rayleigh and SH wave, which we will exploit in our ADFMR measurements. Again, the comparison with the literature sound velocities given in Tab. 3.2 and with literature displacement calculations (see [40]) shows perfect agreement, which

21 Section 1.5 Summary 17 displacement (a.u.) a Shear horizontal wave 36 YX-LiTaO 3 u y u z u x z ( ) electrical potential (a.u.) b z ( ) Figure 1.7: z-dependence of (a) displacement and (b) electrical potential of a SH wave propagating on 36 YX-LiTaO 3, assuming a metallized surface. λ is the SAW wavelength, z = 0 corresponds to the surface of the crystal. SH wave Rayleigh wave sound velocity v met 4108m/s 3124m/s strains ε xx 0.14i 1 ε yy 0 0 ε zz 0.21i 0.12 ε yz 0.05i 0.07 ε xz ε xy i Table 1.3: Calculated sound velocity and strains for SH and Rayleigh wave on 36 YX- LiTaO 3, assuming a metallized surface. indicates the correctness of the used algorithms. 1.5 Summary In this chapter, the basis for a fully quantitative description of ADFMR has been layed. We have presented a general theory of SAW propagation, which holds for any piezoelectric crystal and arbitrary crystal cut and which allows to calculate the mechanical displacement, the electrical potential and the strains induced by the SAW. The presented theory was applied to YZ-LiNbO 3 and 36 YX-LiTaO 3 and the differing

22 18 Chapter 1 Surface acoustic waves on piezoelectric substrates a ε xx b ε zz z y Figure 1.8: Three-dimensional plot of the dominant strain components induced by a Rayleigh wave on 36 YX-LiTaO 3. The propagation direction of the SAW is along the x-axis, z is the surface normal. The color code shows the real part of (a) ε xx and (b) ε zz, where red and blue color denotes positive and negative strain, respectively. The scaling of the color code is the same for both figures. Similar to Fig. 1.1, the displacement has been exaggerated strongly. x character of Rayleigh and shear-horizontal wave as well as the influence of the surface boundary condition free or metallized surface was demonstrated. Comparing the calculation results to literature values showed the expected agreement and proved the reliability of the presented theory and its implementation in Mathematica.

23 Section 1.5 Summary 19 a ε xx b ε zz z c y ε xy x Figure 1.9: Three-dimensional plot of the dominant strain components induced by a shearhorizontal wave on 36 YX-LiTaO 3. The propagation direction of the SAW is along the x-axis, z is the surface normal. The color code shows the real part of (a) ε xx, (b) ε zz and (c) ε xy, where red and blue color denotes positive and negative strain, respectively. The scaling of the color code is the same for the three figures. Similar to Fig. 1.1, the displacement has been exaggerated strongly.

24 20 Chapter 1 Surface acoustic waves on piezoelectric substrates

25 Chapter 2 Modelling of acoustically driven FMR In this chapter, three ways of describing acoustically driven FMR are presented. The first approach is based on the Landau-Lifshitz-Gilbert equation of motion for the magnetization in a ferromagnet, introduced by L. Landau and E. Lifshitz in 1935 [41] and modified by T. L. Gilbert in 2004 [42]. The second model extends this approach and takes into account the backaction of the magnetization precession in the ferromagnet on the SAW, which allows to calculate the change in SAW transmission due to ADFMR. Both models have been developed by L. Dreher and M. Weiler [16,17,43]. The discussion of these models in Secs. 2.1 and 2.2 follows [16] and [17]. The third approach to ADFMR combines the SAW propagation theory presented in Chap. 1 with the methods used in the backaction model in order to describe the magnetization dynamics in the ferromagnet coupled to the acoustic wave propagating in the ferromagnet/piezoelectric crystal hybrid. In this way, the damping effect of ADFMR on the SAW can be fully accounted for. While the LLG model is comparatively simple but instructive to show the physics of ADFMR, the backaction model allows a more detailed description of the transmitted acoustic wave and is therefore more powerful than the LLG approach. The SAW model is an extension of the backaction model. It enlarges its range of validity and describes the full hybrid system consisting of FM and piezoelectric crystal, but is numerically expensive. For this reason, it is useful to consider all three presented approaches when modelling ADFMR. 21

26 22 Chapter 2 Modelling of acoustically driven FMR 2.1 Landau-Lifshitz-Gilbert approach Landau-Lifshitz-Gilbert equation and free-enthalpy density For a phenomenological description of acoustically driven FMR, we start from the wellknown LLG equation [41, 42], which is the equation of motion for the magnetization in a ferromagnet (FM) and widely used in magnetization dynamics: t m = γm µ 0 H eff +am t m. (2.1) Here, m = M M s denotes the magnetization vector normalized to the saturation magnetization M s, and γ and a are the gyromagnetic ratio and a phenomenological damping parameter, respectively. H eff is the effective magnetic field, which exerts a force on the spins in the FM and in this way excites the magnetization precession. In the following, we derive this effective magnetic field H eff. We begin by writing down the Gibbs free-energy density in the FM, considering elastic and magnetoelastic contributions. In absence of elastic strains, the Gibbs freeenergy density of the FM normalized to the saturation magnetization M s can be calculated using a macro-spin approach 1 : G = µ 0 H m+b d m 2 z +B u (m u) 2 µ 0 H ex m. (2.2) Here, H denotesthestaticexternal magneticfield, B d represents theshapeanisotropy of the ferromagnetic thin film, and B u is the uniaxial in-plane anisotropy along the unit vector u [17,43]. µ 0 H ex describes the exchange interaction between the electron spins which can be written as µ 0 H ex = D s m with the exchange stiffness D s and the Laplacian operator = x y + 2 z [19,45]. In the following, we neglect the exchange interaction as its contribution to the free-enthalpy density can be shown to be small compared to the other terms in Eq. (2.2) (see Sect ). The magnetoelastic contribution to the free-energy density is given by 2 [46] G me = b 1 ε ij m i m j, (2.3) with the magnetoelastic coupling constant b 1 and the strain tensor ε ij = 1 2 ( u i x j + u j x i ), 1 The macro-spin model describes a single-domain approximation of the FM, where the electron spins are assumed to all be aligned in parallel due to ferromagnetic exchange interaction(cf. [44]). 2 SimilartoEq.(2.2), G me hasbeennormalizedtothesaturationmagnetizationm s, sothat[g] = T.

27 Section 2.1 Landau-Lifshitz-Gilbert approach 23 i,j {x,y,z}. Using the thermodynamic relation µ 0 H eff = m G tot, (2.4) withg tot = G+G me thesumofallrelevantcontributionstothefree-energydensity, we cancalculate µ 0 H eff asafunctionofthe strainsε ij and themagnetizationcomponents m i Magnetization precession and virtual driving field Without a (radio-frequency) driving field, the magnetization is oriented in a way that G tot is minimized. In polar coordinates, this equilibrium direction m 0 is characterized by two angles φ 0 and θ 0. In the dynamic case, i.e. with a driving field, m precesses around its equilibrium direction. Throughout this thesis, we only consider small deviations from m 0 [17]. It is convenient to define a new coordinate system (x 1,x 2,x 3 ), in which m 0 is along the x 3 -axis and the x 2 -component of m lies in the film plane, as indicated in Fig. 2.1 [17]. Then, m can be written as: m = }{{} m 0 + m 1 m 2 0 +O(m 2 1,m 2 2). (2.5) Thetransformationmatrixrelatingthetwoframesofreference(x,y,z)and(x 1,x 2,x 3 ) can be found in [17]. We expand m G and m G me in terms of (m 1,m 2 ), considering that G m 1 m=m0 = G m 2 m=m0 = 0 by definition of m 0. This leads to G 11 m 1 +G 12 m 2 µ 0 H eff = G 12 m 1 +G 22 m 2 G 3 G me 1 G me 2 G me 3, (2.6) where we use the abbreviations G i = mi G m=m0 and G ij = mi mj G m=m0. Similar to Eq. (1.11), we make a plane-wave ansatz for the transverse magnetization components, m i = m 0 i exp[i(kx ωt)], i {1,2}, with angular frequency ω and wave

28 24 Chapter 2 Modelling of acoustically driven FMR a z H b z m 3 1 m 2 m 1 Θ 0 β m 0 y 3 m 0 Θ 0 2 y m α φ 0 φ 0 x k SAW Nickel 1 x k SAW Figure 2.1: (a)definitionoftheanglesα, β,θ 0 andφ 0 : αandβ parametrizetheorientation of the external magnetic field µ 0 H, and θ 0 and φ 0 denote the equilibrium direction of the magnetization, m 0. (x,y,z) is the sample coordinate system where x, y and z are along the SAW propagation direction, in the film plane and normal to the film plane, respectively. Based on m 0, we define a second frame of reference, (x 1,x 2,x 3 ), as indicated in (b). Here, x 3 is along m 0, x 2 lies in the film plane and x 1 is perpendicular to x 2 and x 3 so that (x 1,x 2,x 3 ) is right-handed. Panel (b) has been taken from [17]. number k. With this ansatz and the linearized µ 0 H eff from Eq. (2.6), the Landau- Lifshitz-Gilbert equation (2.1) can be written in matrix form. As m 3 1, we hereby only regard the transverse magnetization components: ( G11 G 3 iωα γ G 12 iω γ G 12 + iω γ G 22 G 3 iωα γ )( m 1 m 2 ) = µ 0 ( h 1 h 2 ), (2.7) where µ 0 h i = G d i = mi G d m=m0 are the components of the effective driving field 3 3 Please note that by definition the effective driving field vector h = (h 1,h 2 ) T has only two components, in contrast to m or H.

29 Section 2.1 Landau-Lifshitz-Gilbert approach 25 h: µ 0 h 1 = 2b 1 sinθ 0 cosθ 0 [ε xx cos 2 φ 0 +ε yy sin 2 φ 0 ε zz ] 2b 1 [(ε xz cosφ 0 +ε yz sinφ 0 )cos(2θ 0 )+2ε xy sinθ 0 cosθ 0 sinφ 0 cosφ 0 ] (2.8) µ 0 h 2 = 2b 1 sinθ 0 sinφ 0 cosφ 0 [ε xx ε yy ] 2b 1 [cosθ 0 (ε yz cosφ 0 ε xz sinφ 0 )+ε xy sinθ 0 cos(2φ 0 )]. (2.9) While in conventional FMR the driving field h is a real electromagnetic field which is applied to the FM, in ADFMR, h is purely virtual and is introduced to describe the magnetoelastic interaction between the radio-frequency elastic strain and the magnetization in the FM. Solving Eq. (2.7) for (m 1,m 2 ) T gives ( m 1 m 2 with the Polder susceptibility tensor [45] ) ( = χ h 1 h 2 ) (2.10) χ = ( G11 G 3 iωα γ G 12 iω γ G 12 + iω γ G 22 G 3 iωα γ ) 1. (2.11) As we can see from Eq. (2.10), the magnetization precession amplitude is the product of the virtual driving field h induced by the elastic strains in the FM and the Polder susceptibility tensor, which depends on the material parameters γ, α, B d, B u and the static external magnetic field H. This tensor reflects the static properties of the ferromagnetic thin film as is does not depend on any time-dependent quantity. The virtual driving h field describes the dynamics of ADFMR and shows unlike conventional FMR a pronounced m 0 -dependence, which changes with the applied strain components. To illustrate this, Fig. 2.2 shows polar plots of µ 0 h for different applied strains ε ij and different orientations (θ 0,φ 0 ) of the equilibrium magnetization m 0 with respect to the SAW propagation direction. Regarding the first row in Fig. 2.2 (denoted as IP configuration), we see that for a pure longitudinal strain ε xx, h is maximum at φ 0 = ±45, but vanishes at 0 and ±90, resulting in a characteristic four-fold butterfly shape. For a Rayleigh wave, ε xx dominates (see Tab. 1.2), so we expect an ADFMR angle-dependence similar to

30 26 Chapter 2 Modelling of acoustically driven FMR ε xx 0 ε xy 0 ε xz 0 a y b y c y φ 0 φ 0 φ 0 IP' θ 0 =π/2 x x x 50µT d OOP1' φ 0 =0 z e z f z θ 0 θ 0 θ 0 x x x OOP2' φ 0 =π/4 g v=(x+y)/ 2 z z z θ h i 0 θ 0 θ 0 v v v Figure 2.2: Polar plot of µ 0 h for different orientations and strain components (from [17]). In Panel (a) to (c), m 0 lies in the film plane (θ 0 = π/2) at an angle of φ 0 to the SAW propagation direction. Panel (d) to (f) show µ 0 h for a θ 0 -rotation in the plane φ 0 = 0, which is perpendicular to the film plane. In Panel (g) to (i), finally, m and the SAW propagation direction x enclose an angle of π/4. For vanishing anisotropy (B u = B d = 0), and thus H m 0, these rotation planes correspond to the IP, OOP1 and OOP2 configuration which will be introduced in Sect. 3.2, so they have been labeled IP, OOP, and OOP2. The plotted µ 0 h has been calculated using Eqs. (2.8) and (2.9), setting all strain components to zero except from the one indicated ((a), (d) and (g): ε xx 0, (b), (e) and (h): ε xy 0, (c), (f) and (i): ε xz 0). The plotted scale bar holds for b 1 = 25T and the non-vanishing strain component ε ij = 10 6.

31 Section 2.1 Landau-Lifshitz-Gilbert approach 27 Fig. 2.2a. For a pure ε xy strain (see Fig. 2.2b), the characteristic angle-dependence is rotated by 45, so that h reaches its maximum at φ 0 = 0 and ±90 and is zero at ±45. As ε xy is the dominant strain component for a SH wave on 36 YX-LiTaO 3 (see Tab. 1.3), the corresponding ADFMR signature should reflect an angle-dependence similar to Fig. 2.2b. InChap.3, wewillpresenttheexperimentalresultsforrayleighandshwavedriven FMR, and we will show that the observed angle-dependence matches the theoretical considerations. In conventional FMR experiments, in contrast, h does not depend on the orientation of the static magnetic field. Therefore, a polar plot of h similar to Fig. 2.2 would simply yield a circle Absorbed radio-frequency power It is difficult to directly measure magnetization precession via microwave-based equipment. Therefore, we calculate the microwave power which is absorbed by ADFMR, as this can easily be detected by means of network analysis. According to [47], the absorbed power P abs in conventional FMR can be written as P abs = ωµ 0 2 V 0 Im [ ( (h ) h 1,ext 1,ext,h 2,ext χ h 2,ext )] dv, (2.12) where V 0 is the volume of the ferromagnetic film and h ext denotes the external AC magnetic driving field. We adapt this formula to ADFMR by identifying h ext with the virtual driving field h, which is due to magnetoelastic coupling. In Fig. 2.3, P abs is plotted against magnitude and orientation of the external static magnetic field µ 0 H, together with the virtual driving field µ 0 h and the susceptibility χ. While in conventional FMR, µ 0 h is independent of the static magnetic field orientation α and therefore the angle-dependence of P abs resembles the one of the susceptibility χ, in ADFMR µ 0 h depends on the magnetization direction m as well as on the applied strains (cf. Sect ), so that thecharacteristic angle-dependence of P abs in ADFMR looks qualitatively different from conventional FMR.

32 28 Chapter 2 Modelling of acoustically driven FMR Figure 2.3: H-dependence of (a) virtual driving field µ 0 h, (b) susceptibility χ and (c) absorbed power P abs, as calculated with Eqs. (2.8), (2.11) and (2.12). H lies in the film plane θ = 0, at an angle of φ to the SAW propagation direction x. The frequency was set to ν = 2.24GHz, the anisotropy parameters are B d = 400mT, B u = 2.5mT and u = x. The saturation magnetization M s and the gyromagnetic ratio γ have been taken from Tab The strain was assumedto be purely longitudinal, i.e. only ε xx 0, and the damping parameter was taken as a = Effects of shear strains on ADFMR symmetry Assuming a strain tensor ε ij with ε xz = ε yz = 0, the ADFMR signature is symmetric with respect to the external field µ 0 H. This means that a rotation µ 0 H by π does not change the magnetization precession amplitudes and the absorbed power. Adding one of the off-diagonal strain components ε xz or ε yz, however, breaks this symmetry. This is illustrated in Fig. 2.4, where the power absorption calculated with the LLG model is shown for different strains ε ij. To study the effect of ε xz or ε yz strains on the ADFMR symmetry theoretically, we first regard Eq. (2.2). Assuming that, for a given set of parameters B u, B d and u, the free-enthalpy density G is minimized by some m 0 at a certain field H 0, then m 0 minimizes G at H = H 0, as G(H,m 0 ) = G( H, m 0 ). Thus, an inversion of the external field direction also reverses the equilibrium magnetization direction m 0. We now introduce a transformation T, which describes the inversion of the equilibrium magnetization orientation m 0 : T : m 0 m 0. (2.13)

33 Section 2.1 Landau-Lifshitz-Gilbert approach 29 Figure 2.4: Angle-dependence of P abs for different strains ε ij. The simulation parameters and the angle definition is the same as in Fig. 2.3, the strains have been chosen as following: (a) only ε xx 0, (b) ε xx 0 and ε xz = iε xx /3 and (c) ε xx 0 and ε yz = iε xx /3. P abs is plotted in arbitrary units and normalized to its maximum. Regarding the angles θ 0 and φ 0, an inversion of m 0 does not affect θ 0, but shifts φ 0 by π. Thus, T : θ 0 θ 0 φ 0 φ 0 +π. (2.14) In anticipation of the next paragraph, we give the corresponding transformations of sine and cosine of θ 0 and φ 0 : T : sin(θ 0 ) sin(θ 0 ) cos(θ 0 ) cos(θ 0 ) sin(φ 0 ) sin(φ 0 ) (2.15) cos(φ 0 ) cos(φ 0 ). Regarding Eq. (2.8) and (2.9), we see that for a purely diagonal strain tensor ε ij, i.e. ε yz = ε xz = ε xy = 0, the virtual driving field h is given by µ 0 h 1 sinθ 0 cosθ 0 [ε xx cos 2 φ 0 + ε yy sin 2 φ 0 ε zz ] and µ 0 h 2 sinθ 0 sinφ 0 cosφ 0 [ε xx ε yy ]. Obviously, h 1 and h 2 are invariant under the transformation T, as all factors containing φ 0 are of the form sin 2 (φ 0 ), cos 2 (φ 0 ) or sin(φ 0 )cos(φ 0 ), which do not change sign under T. As the susceptibility tensor χ is invariant under T, too, this also holds for the magnetization (m 1,m 2 ) T = χh and the absorbed power P abs h χh. As one can easily see from Eq. (2.8) and (2.9), the T-invariance of (m 1,m 2 ) T and P abs also

34 30 Chapter 2 Modelling of acoustically driven FMR remains for a non-zero ε xy strain component. When introducing an additional ε xz or ε yz strain, however, also odd-power terms of sin(φ 0 ) and cos(φ 0 ) appear, which breaks the symmetry of (m 1,m 2 ) T and P abs with respect to an inversion of m Backaction of ADFMR on the SAW In the LLG model introduced in the previous paragraph, we have calculated the magnetization precession starting from the free-enthalpy density for the FM. Moreover, we have derived a formula for the power absorption due to ADFMR which can be compared to the experiment. In this section, the backaction of ADFMR on the acoustic wave will be examined. We thereby aim at the calculation of the complex S 21 -parameter, which describes damping and phase shift of the acoustic wave. As in Chap. 1, we start with the equation of motion in a solid, Eq. (1.1), to describe the elastic displacement in the ferromagnetic film: ρ 2 u i t 2 = j σ ij x j, i,j {1,2,3}. (2.16) Here, u is the displacement and σ ij the stress tensor in the FM. In this section, we neglect the neighboring piezoelectric crystal and its coupling to the ferromagnetic film. In its most general form, the stress tensor σ ij can be written as [25,48] σ ik = W ε ik, (2.17) where W is the elastic energy density in the FM. In absence of magnetoelastic interactions, W is given by [26] W el = 1 2 ε ij c ijkl ε kl. (2.18) i,j,k,l The magnetoelastic contribution to the elastic energy density has been derived in

35 Section 2.2 Backaction of ADFMR on the SAW 31 Sect (cf. Eq. (2.3)) and is 4 W me = G me M s. (2.19) As above, we make a plane-wave ansatz for the elastic displacement and the transverse magnetization components, u x,y,z = u 0 x,y,z exp(ikx iωt) and m 1,2 = m 0 1,2exp(ikx iωt), which is substituted into Eq. (2.16). Using W = W el + W me and Eqs. (2.18) and (2.19), we get ρω 2 u x = c 11 k 2 u x +2ib 1 M s ksinθ 0 cosφ 0 [sinφ 0 m 2 cosθ 0 cosφ 0 m 1 ] (2.20) ρω 2 u y = c 44 k 2 u y 2ib 1 M s ksinθ 0 [2sinφ 0 cosφ 0 cosθ 0 m 1 +cos(2φ 0 )m 2 ](2.21) ρω 2 u z = c 44 k 2 u z +2ib 1 M s k[sinφ 0 cosθ 0 m 2 cos(2θ 0 )cosφ 0 m 1 ]. (2.22) As we have assumed an elastically isotropic ferromagnetic film, the stiffness tensor has only two independent components, c 11 and c 44. These are related to the commonly used Lamé constants λ and µ by c 11 = λ+2µ and c 44 = µ [37]. The linearized LLG equation (2.10) and Eqs. (2.20) to (2.22) form a system of five linear equations for the unknown quantities u x, u y, u z, m 1 and m 2. The determinant of this system of equations has to vanish to get a non-trivial solution, which allows to determine the wave number k. To be able to proceed analytically, we neglect transverse and normal displacement component in the following (i.e., assume u y = u z = 0) and consider a purely longitudinal acoustic wave. Substituting Eqs. (2.8) to (2.11) into Eq. (2.20), we get [ ω 2 v 2 l ( 1 Fb2 { } ) ] 1 vl 2µ χ11 w1 2 +χ 22 w2 2 (χ 12 +χ 21 )w 1 w 2 k 2 u x = 0 (2.23) 0ρ with the abbreviations w 1 = 2sinθ 0 cosθ 0 cos 2 φ 0, w 2 = 2sinθ 0 cosφ 0 sinφ 0 and v l = c11 /ρ. Here, we have introduced a filling factor F < 1, which accounts for the fact that only a small fraction of the volume in which the SAW propagates is ferromagnetic and which therefore reduces the coupling between FMR and SAW. As the penetration depth of the SAW is of the order of the wavelength λ, F is of the order of t Ni /λ, where t Ni is the thickness of the ferromagnetic film. 4 G and G me from Chap. 1 have been normalized to the saturation magnetization M s. As W me is meant to be an energy density, i.e. energy per volume, an additional factor M s appears in Eq. (2.19).

36 32 Chapter 2 Modelling of acoustically driven FMR Without magnetoelastic coupling, the wave number is k = k 0 = ω v l. Assuming that magnetoelastic coupling involves only small changes in the elastic wave mode, we write k = k 0 + k, with k k 0, and neglect higher orders of k in Eq. (2.23). Then, we get k = F ωb2 1 2v 3 l µ 0ρ {χ 11w 2 1 +χ 22 w 2 2 (χ 12 +χ 21 )w 1 w 2 }. (2.24) Substituting Eq. (2.24) into the plane-wave ansatz u x = u 0 xexp(ikx iωt) yields the complex transmission coefficient S 21 := u x(x = l Ni ) u x (x = 0) = ei kl Ni, (2.25) where l Ni is the length of the ferromagnetic film. S 21 describes damping and phase shift of the acoustic wave due to ADFMR and is normalized so that S 21 = 1 off resonance. It can be shown [17] that for small k and pure longitudinal strain (i.e. ε xx 0 only) the backaction model is equivalent to the Landau-Lifshitz-Gilbert approach introduced in Sect Phase changes of the order of π/2, as they are observed in the experiment (see Secs. 3.4 and 3.5), are beyond the limits of the backaction model and cannot be reproduced. This is due to the first-order expansion of Eq. (2.23), which is necessary to get an analytical expression for k. This constraint could be avoided using numerical calculations instead of an analytical approach, but as the consistency of LLG and backaction model only holds for small k, as shown in [17], even an exact solution of Eq. (2.23) would not extend the range of validity of the backaction model significantly. 2.3 Surface acoustic wave model In the backaction model presented in Sect. 2.2, we have studied the effects of magnetoelastic coupling on the elastic wave in a ferromagnetic thin film and calculated the complex S 21 -parameter. The interaction of the elastic wave in the FM with the SAW propagating on the piezoelectric crystal, however, was considered only via a phenomenological filling factor F < 1. Moreover, the derivation of the complex transmission in the backaction model required the assumption of low damping/dispersion,

37 Section 2.3 Surface acoustic wave model 33 t Ni Ferromagnetic thin film z x=0 x=l Ni x LiNbO 3 k SAW Figure 2.5: Schematic of the FM/piezoelectric crystal hybrid, as modelled in Chap t Ni and l Ni denote the thickness and the length of the ferromagnetic film, respectively. which is not satisfied in practice. In this section, we want to extend the backaction model to achieve a quantitative description of the elastic displacement in both piezoelectric crystal and ferromagnetic thin film including the magnetization dynamics in the FM Equations of motion and plane-wave ansatz As illustrated in Fig. 2.5, we consider a piezoelectric crystal covered with a thin ferromagnetic metallic layer, which we assume to be isotropic concerning its elastic properties 5 and perfectly conducting. For clarity of notation, in the following u and v denote the elastic displacement in the piezoelectric crystal and the FM, respectively. All other quantities, like γ, σ ij and the stiffness constants c ijkl are marked with a Ni when referring to the ferromagnetic film; otherwise, they are related to the piezoelectric. The wave number in the piezoelectric crystal and in the ferromagnetic film is the same, and therefore the symbols β and k are equivalent in the following. The equations of motion for the piezoelectric have been derived in Sect. 1.1 and are given by Eqs. (1.6) and (1.7). For the FM, the displacement v obeys Eq. (2.16) from Sect Together with Eqs. (2.17) to (2.19), we get 2 v i ρ Ni t = c Ni 2 j,k,l ijkl 2 v k x j x l + j σ me ij x j, i {1,2,3} (2.26) with σ me ij = (Gme M s ) ε ij. (2.27) As inchap. 1, we use aplane-wave ansatzforu, v andφwith propagationdirection 5 In the experiment, we use nickel thin films with a thickness of 50nm, which are polycrystalline and therefore macroscopically isotropic regarding their elastic properties.

38 34 Chapter 2 Modelling of acoustically driven FMR x and exponential z-dependence: u(x,t) = u 0 exp(iγz)exp(iβx iωt) (2.28) v(x,t) = v 0 exp(iγ Ni z)exp(iβx iωt) (2.29) φ(x,t) = φ 0 exp(iγz)exp(iβx iωt). (2.30) Duetothecoupling ofthewave inthepiezoelectric crystal andthefmatz = 0, the wave number β is the same in both media, as already mentioned. The z-dependence of u and v may be different, which is reflected by two independent parameters γ and γ Ni. As above, we set the additional constraint Imγ < 0 to guarantee exponential decay of the acoustic wave in the piezoelectric crystal. This condition is not necessary for γ Ni due to the finite thickness of the FM. Substituting the ansatz (2.28) to (2.30) into the equations of motion (1.6), (1.7) and (2.26) yields two homogeneous systems of linear equations whose determinants are polynomials of 8 th degree in γ and 6 th degree in γ Ni, respectively. Because of Imγ < 0, we get four solutions for γ and six solutions for γ Ni (cf. Sect. 1.3), which we denote γ α and γni. α Interface and boundary conditions The acoustic wave in the layered crystal system is a superposition of the fundamental solutions, given by Eqs. (2.28) to (2.30) and the γ α and γ α Ni calculated above: u tot (x,t) = v tot (x,t) = φ tot (x,t) = 4 A α u α (x,t) (2.31) α=1 6 B α v α (x,t) (2.32) α=1 4 A α φ α (x,t). (2.33) α=1 The superposition coefficients A α and B α have to be chosen such that the appropriate interface and boundary conditions are satisfied. First, the elastic displacement has to be continuous at the interface z = 0: u tot = v tot at z = 0. (2.34)

39 Section 2.3 Surface acoustic wave model 35 As we have assumed the ferromagnetic film to be perfectly conducting, the electrical potential in the piezoelectric crystal vanishes at the interface: φ tot = 0 at z = 0. (2.35) Next, the forces acting on an infinitesimal volume element at the interface have to sum up to zero. Therefore, the normal component of the stress tensor σ has to be continuous: σ iz = σ Ni iz at z = 0, i {x,y,z}. (2.36) Finally, at the surface, the normal component of the stress tensor vanishes as there are no forces normal to the surface (cf. Chap. 1): σ Ni iz = 0 at z = t Ni, i {x,y,z}, (2.37) where t Ni is the thickness of the ferromagnetic thin film. In total, we have ten equations which are linear with respect to the unknown coefficients A α and B α. Again, this yields a determinant D 2 which has to be zero to get a non-vanishing SAW solution 6. Using an iterative algorithm to find the correct wave number, β turns out to be complex now, with a small negative imaginary part which describes the damping of the SAW by ADFMR Calculation results In this section, some exemplary calculation results are shown which were obtained with the SAW model. First, the damping of the SAW is calculated for a Rayleigh wave propagating on YZ-LiNbO 3, covered with a t Ni = 50nm thick and l Ni = 0.57mm long nickel film. The material parameters for LiNbO 3 have been taken from Tab. 1.1, those of nickel are listed in Tab For simplicity, we neglect uniaxial anisotropy, i.e. set B u = 0, and assume B d = µ 0 M s /2 = 230mT [46]. The frequency was chosen ν = 1.55GHz, and the damping parameter was taken as a = 0.1. The magnetoelastic coupling constant was assumed to be 14 T and thus was inten- 6 While an approach similar to Chap. 1, i.e. solving D 2(β) = 0 numerically, is mathematically correct, it is not the best way to determine the A α and B α. Instead, we present an alternative algorithm in Appendix A which yields more exact and numerically stable results and therefore has been used for the calculations shown below.

40 36 Chapter 2 Modelling of acoustically driven FMR Density ρ 8900 kg/m 3 Lamé constants λ Ni N/m 2 µ Ni N/m 2 Saturation magnetization M s 370 ka/m Magnetostriction constant λ s Gyromagnetic ratio γ 2.185µ B / Table 2.1: Material constants for the polycrystalline nickel film, taken from [51] (density), [52] (Lamé constants), [53] (saturation magnetization), [54] (magnetostriction constant) and [55] (gyromagnetic ratio). The Lamé constants have been calculated from Young s modulus E = N/m 2 and the shear modulus G = N/m 2 given in [52], using the relations G = µ and E = µ(3λ+2µ)/(λ+µ) from [37]. tionally chosen smaller than the literature value, which is given by b 1 = 3λ s µ Ni = 23T [46] with the isotropic magnetostriction constant λ s and the Lamé constant µ Ni from Tab The reason is that due to pinning of the electron spin at the surface [18, 49, 50] and the ferromagnetic exchange interaction between the spins, the magnetization precession amplitude is effectively reduced. To treat this phenomenon exactly would require solving a coupled system of partial equations instead of algebraic equations, which is much more complex and numerically costly. To avoid this, the effective reduction in magnetization motion was taken into account by introducing an effective magnetoelastic coupling constant b 1 = 14T. In Appendix B, an estimate of the pinning effects on the magnetization precession amplitude is given. Figure 2.6 shows the displacement magnitude and phase, u and arg(u), of the SAW having crossed the nickel film, i.e. at x = l Ni. This is plotted for two different external magnetic fields: The solid lines show the simulated displacement components for µ 0 H = 150mT (off resonance), while the dashed lines correspond to µ 0 H = 7mT, which is the resonance field at the chosen frequency. The plotted magnitude data are normalized to the maximum displacement at the surface and the beginning of the nickel film, u z (x = 0,z = 0). The angle between the external magnetic field and the SAW propagation direction was α = 45, as for this orientation, the driving field is maximum for a dominant ε xx strain (see Fig. 2.2). We can see that off resonance the SAW is not damped at all, as u z (x = l Ni,z = 0) = u z (0,0). In resonance, we observe a strong decrease in amplitude for all

41 Section 2.3 Surface acoustic wave model 37 normalized displacement u i / u z (x=0) a off resonance in resonance u z u x u y z ( ) displacement phase 2 3 /2 /2 0 b off resonance in resonance Arg(u x ) Arg(u z ) z ( ) Figure 2.6: (a) Magnitude and (b) phase of the displacement u at x = l Ni = 1.2mm as a function of z for a Rayleigh wave on YZ-LiNbO 3 (ν = 1.55GHz, α = 45 ). The solid and dashed lines show the cases µ 0 H = 150mT (off resonance) and µ 0 H = 5mT (in resonance), respectively. As u y = 0 within numerical errors, the phase is undefined and therefore not shown in (b). Re( ) (10 3 m -1 ) resp. Im( ) (10 3 m -1 ) a Im( ) (10 3 m -1 ) S 21 =0.07 v/v=0.1% Re( ) (10 3 m -1 ) µ 0 H (mt) driving field (a.u.) magnetization amplitude (a.u.) b c µ 0 h 2 µ 0 h 1 m 1 m µ 0 H (mt) Figure 2.7: Field-dependence of (a) wave number β, (b) virtual driving field h and (c) magnetization precession amplitude (m 1,m 2 ), calculated at x = l Ni = 1.2 mm. The calculation parameters and the substrate are the same as in Fig displacement components, and a phase shift of about π/4, which is uniform regarding z. The strong z-dependence of arg(u x ) around z λ/6 is due to the zero crossing of u x at this point.

42 38 Chapter 2 Modelling of acoustically driven FMR Figure 2.7a shows the field-dependent variation of the wave number β for the same configuration as above. The change of Re(β), which is proportional to the phase of the wave, indicates a variation of the propagation velocity of the wave, which is at most v/v = β/β 0.1%. Im(β) corresponds to the damping of the wave via S 21 exp( Im(β)l Ni ), yielding S 21 7% at µ 0 H res = 5mT. In Fig. 2.7b and c, the virtual driving field µ 0 h and the magnetization amplitude (m 1,m 2 ) are plotted as a function of the external field µ 0 H. Obviously, the fielddependence of h is strongly asymmetric, while (m 1,m 2 ) is nearly symmetric and shows clear peaks at the resonance fields, with m 1 being the dominant component. The discontinuity of β, h and m at µ 0 H = 0 is due to the fact that in this model the equilibrium magnetization m 0 is constant in magnitudebut abruptly changes sign at µ 0 H = 0. At µ 0 H = 0, m 0 is undefined. The asymmetry of β, h and m with respect to the external field µ 0 H results from theshearstrainε xz inducedbytherayleighwaveandhasbeendiscussedinsect Figure 2.8 shows the characteristic angle-dependence of the wave number β in ADFMR (on YZ-LiNbO 3 ), using the parameters given above. The driving field and therefore also the ADFMR signal is strongest at α = ±45, whereas it vanishes at 0 and ±90 (cf. Sect ). To study the effects of the SAW type on the ADFMR signature, the angle-dependence of β has been calculated for a 50nm thin nickel film on 36 YX-LiTaO 3 at 780MHz and 1.04GHz, which corresponds to the 5 th harmonic of Rayleigh and SH wave, respectively. The simulation parameters were b 1 = 8T and a = 0.16 (Rayleigh wave) resp. 0.5 (SH wave), with B u = 0 and B d = 230mT as above. The results are plottedinfigs.2.9and2.10. FortheRayleighwave, wenoticeamaximumchangeinβ at α = ±45 and a vanishing β at 0 and ±90. For the SH wave, this characteristic signature is shifted by 45, as β is maximum at 0 and ±90, but vanishes for α = ±45. This confirms the theoretical considerations from Sect , where the characteristicadfmrsignaturewaspredictedbasedontheangle-dependenceofµ 0 h. 2.4 Summary In this chapter, we have introduced three different approaches to model ADFMR. In the first method, called LLG approach, we derived an effective driving field µ 0 h from the magnetoelastic contribution to the free-enthalpy density G tot. Substituting µ 0 h into the Landau-Lifshitz-Gilbert equation, the magnetization precession and the

43 Section 2.4 Summary 39 Figure 2.8: Angle-dependence of (a) real and (b) imaginary part of the wave number β, calculated with the SAW model. µ 0 H lies in the film plane, α denotes the angle between µ 0 H and k SAW (cf. Fig. 2.1a). The parameters used for the calculation are given in the text. The SAW is assumed to be a Rayleigh wave, propagating on YZ-LiNbO 3. absorbed power P abs can be calculated. In the backaction model, the effects of the magnetization precession on the acoustic wave in the FM are considered. To this end, a system of coupled equations has been established, describing the magnetization components m 1 and m 2 as well as the elastic displacement u induced by the acoustic wave. The self-consistent solution of this coupled system yields the magnetization precession and the corresponding change in u, which can be expressed in the complex transmission parameter S 21. The third model combines the backaction model with the SAW propagation theory presented in Chap. 1. Thus, not only the acoustic wave in the ferromagnetic thin film is considered, but the combined system of a SAW on the crystal, the accompanying acoustic wave in the FM and the induced magnetization precession is solved selfconsistently. The particular type of SAW and the backaction of the magnetization precession on the SAW are inherently considered. From a theory perspective, the SAW-FM interaction thus is described on different levels of sophistication in the three models. What remains to be addressed in future work is a quantitative treatment of the SAW excitation using interdigital transducers.

44 40 Chapter 2 Modelling of acoustically driven FMR Figure 2.9: Angle-dependence of (a) real and (b) imaginary part of the wave number β, calculated with the SAW model. µ 0 H lies in the film plane, α denotes the angle between µ 0 H and k SAW (cf. Fig. 2.1a). The parameters used for the calculation are given in the text. The SAW is assumed to be a Rayleigh wave, propagating on 36 YX-LiTaO 3. Figure 2.10: Angle-dependence of (a) real and (b) imaginary part of the wave number β, calculated with the SAW model. µ 0 H lies in the film plane, α denotes the angle between µ 0 H and k SAW (cf. Fig. 2.1a). The parameters used for the calculation are given in the text. The SAW is assumed to be a shearhorizontal wave, propagating on 36 YX-LiTaO 3.

45 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory Having derived three theoretical approaches to model ADFMR in the previous chapter, we now turn to the experiment. In the following, we first introduce the samples and the experimental setup. Then a physical interpretation of the measured reflection and transmission coefficient is given and the data post-processing is explained. Finally, we present the experimental data and compare them to the calculation results obtained using the models from Chap Samples For the acoustically driven FMR (ADFMR) measurements, the Ni/Al-hybrid samples LNOC119, denoted as Sample 1 in the following, and LNO-36Y-14 1 (Sample 2), were used. Sample 1 was fabricated by H. Söde [56] and is shown in [57]. Sample 2 was fabricated as a part of this thesis; a photograph of Sample 2 is given in Fig Both devices consist of a 1mm thick, one-side polished piezoelectric crystal, Y-cut LiNbO 3 resp. 36 -rotated Y-cut LiTaO 3, on which a pair of aluminium interdigital transducers (IDTs) was patterned using optical lithography and e-beam evaporation. The width 1 This sample has been chosen for the ADFMR measurements as the quality of the lithographically defined IDTs is very good, so that not only the fundamental SAW frequency but also higher harmonics can be observed (as it will be shown below). 41

46 42 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory a b IDTs Ni film Alignment marker mini-smp connectors 20mm 6mm 10mm LTO-36Y YX-LiTaO 3 crystal Figure 3.1: (a) Photograph of Sample 2, mounted on a chipcarrier which has been designed for ADFMR and acoustic spin pumping measurements. (b) Enlarged view of Sample 2. As the crystal is transparent, the glue used to mount the sample onto the chipcarrier appears as a clear spot. Aluminum, N=30 w IDT =500µm p (20µm) f (5µm) Figure 3.2: Schematic of the IDT design used in this work. f and p are the width of the IDT fingers and the periodicity of the fingers, respectively. N denotes the number of finger pairs of an IDT, and w IDT is the aperture of the IDT, i.e. the length of the IDT fingers. of the IDT fingers is 5µm, with a metallization ratio 2 of r = 2f/p = 50%, leading to a fundamental wavelength of 20µm (see Fig. 3.2) 3. As shown in Fig. 3.3, a t Ni = 50nm thin polycrystalline nickel 4 film of rectangular shape was deposited on the delayline between the IDTs using e-beam evaporation. In Tab. 3.1, the most important geometric dimensions of both samples are listed. The fabrication parameters can be found in Appendix C and in [56, 57], respectively. 2 As indicated in Fig. 3.2, f and p denote the width of a IDT finger and the periodicity of the fingers, respectively. 3 For a detailed explanation of the working principle of IDTs, we refer to [43]. 4 Nickel has a high magnetostriction constant and a comparatively low coercive field, which makes it well suited for ADFMR experiments.

47 Section 3.1 Samples 43 p=20 µm l Ni N=50 w Ni w IDT =500 µm d IDT Figure 3.3: Schematic of the ADFMR samples. w Ni and l Ni denote width and length of the nickel thin film, and d IDT is the distance between the IDT centers. f, p, N and w IDT are defined in Fig t IDT d IDT t Ni l Ni w Ni nm mm nm mm mm Sample Sample Table 3.1: Geometric dimensions of IDTs and nickel films on Sample 1 and Sample 2. In Tab. 3.2, the key parameters of the SAWs propagating on Sample 1 and Sample 2 are summarized from literature [40, 58]. Besides the fundamental SAW mode with a wavelength of 20µm, IDTs also allow the excitation of higher harmonics, depending on the metallization ratio r. Assuming exactly r = 1/2, only each forth harmonic can be observed [59]. It will be shown substrate prop. dir. wave type κ 2 v free v met % m/s m/s Sample 1 Y-cut LiNbO 3 Z Rayleigh Sample rot. Y-cut LiTaO 3 X SH Rayleigh Table 3.2: Literature SAW properties of Sample 1 and Sample 2. κ 2 is the electromechanical coupling factor, v free and v met denote the sound velocity of the respective SAW, assuming a free or metallized surface. Values for κ 2, v free and v met have been taken from [58] (LiNbO 3 ) and [40] (LiTaO 3 ). As v met for LiNbO 3 is not given explicitly in [58], it has been calculated from v free and κ 2 = 2(v met v free )/v free [58].

48 44 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory a VNA rf dipstick gaussmeter DUT rotable electromagnet hall probe power supply cooling water b IDT 1 z β k SAW α y x H Nickel film IDT 2 LiNbO 3 Figure 3.4: (a) Schematic of the ADFMR setup. (b) Enlarged view of the DUT. The mounting orientation of the sample in the rf dipstick can be varied using different adaptor units. Thus, the magnetic field direction relative to the SAW propagation direction can be rotated in different planes. below (see Fig and 3.15) that this is fulfilled quite well in the above samples. 3.2 Experimental setup The setup for the ADFMR measurements has been constructed by M. Weiler [16, 43]. Here, only a short overview of the experimental setup is given; a more detailed description can be found in [43]. The main components of the ADFMR setup are: a radio-frequency (rf) dipstick, which contains the device under test (DUT) and provides connectors for the microwave cables a rotable electromagnet for the generation of the static external magnetic field a vector network analyser (VNA), which is used to measure the transmission and reflection coefficients S ij of the DUT A schematic view of the ADFMR setup is shown in Fig In order to avoid any movement of sample and microwave cables which would influence the high-frequency measurement, the rf dipstick is fixed to the rack. The DUT is mounted at the lower end of the dipstick so that is located between the pole shoes of the electromagnet. The magnet can be rotated by about 200, which allows

49 Section 3.2 Experimental setup 45 to rotate the magnetic field direction with respect to the DUT. The DUT is fixed to a chipcarrier providing two rf mini-smp connectors which are bonded to the IDTs on the DUT. Via semi-rigid coaxial cables in the rf dipstick and SMA connectors at the top end of the dipstick, the DUT is connected to the vector network analyser. The VNA, a Rohde&Schwarz ZVB8, covers a frequency range from 300kHz up to 8GHzand provides a maximum microwave output power of 30dBm. Inorder to avoid thermal drift and nonlinear effects, power was chosen as low as possible, depending on the magnitude and the signal-to-noise ratio of the measured transmission signal (see below). The external static magnetic field is generated by a Oerlikon electromagnet, which is connected to a LakeShore 642 PowerSupply. With a maximum output of 40V and 70A, DC fields up to 1.2T can be generated. The power supply is feedback-controlled by a LakeShore 475 DSP Gaussmeter and a Hall sensor mounted between the pole shoes of the electromagnet. as The reflection and transmission coefficients S ij measured by the VNA are defined S ij = b j a i, (3.1) where a i and b i denote the emitted and detected AC voltage at port i. Usually, S ij is expressed in logarithmic units [57]: S ij (db) = 20log S ij. (3.2) As already mentioned, the electromagnet can be rotated around its vertical axis. In this way, the relative orientation between µ 0 H and k SAW can be varied. The rotation plane is determined by the mounting orientation of the sample in the rf dipstick. To simplify the interpretation of the experimental data, we define three physically interesting rotation planes, as illustrated in Fig. 3.5: In the in-plane configuration, the external field lies in the nickel film plane. In the out-of-plane 1 and out-of-plane 2 configuration, µ 0 H is rotated in a plane perpendicular to the nickel film, with an angle of 0 resp. 45 to k SAW. All measurements were performed at room temperature and standard ambient conditions.

50 46 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory a z IP b OOP1 c OOP2 z z ψ µ 0 H ψ µ 0 H y y y ψ µ 0 H v x k SAW x k SAW π/4 x k SAW Figure 3.5: Measurement configurations employed in the ADFMR experiments (taken from [17]): (a) In the in-plane configuration (IP), the external magnetic field µ 0 H is rotated in the film plane. (b) and (c) In the out-of-plane 1 (OOP1) and out-of-plane 2 (OOP 2) configuration, the external field is rotated in a plane defined by the surface normal z and the vector x resp. v = (x+y)/ 2. IDT 1 EMW (v m/s) SAW (v m/s) IDT 2 SAW triple transit Figure 3.6: Main contributions to the measured transmission coefficient S 21 : Crosstalk of the electromagnetic wave (EMW), SAW pulse and triple transit. 3.3 Data processing SAW transmission Earlier work by M. Weiler [16,43] and C. Heeg [57] showed that the frequencydependent transmission S 21, i.e. the ratio of the voltage detected at IDT2 over the voltage applied to IDT 1, can be described as a superposition of different signals, as indicated in Fig. 3.6: 5 First, the electromagnetic crosstalk, which arises from the fact that only a small fraction of the microwave power applied to IDT1 is converted to a SAW, while the rest is reflected or emitted as a free-space electromagnetic wave. Due to reciprocity [59], this electromagnetic wave is detected by IDT 2 and contributes to the measured 5 In the following, the SAW is always assumed to propagate from IDT1 to IDT2, i.e. IDT1 is the emitter and IDT2 the detector of the SAW.

51 Section 3.3 Data processing 47 transmission signal. The electromagnetic wave propagates with the speed of light c m/s and therefore arrives almost instantaneously, i.e. within about 10ps, at IDT2. Second, the main SAW pulse, which is launched by IDT1, travels with speed of sound v SAW m/s and thus can be observed at IDT2 about t = d IDT v SAW 0.4µs after it has been launched. Third, as an IDT also acts as a (bad) reflector, the SAW pulse is reflected at IDT2, travels back to IDT1, is reflected there again and arrives at IDT2 at t 3t. This secondary SAW pulse is called triple transit. Apart from these, there are several more spurious signals due to reflections of the SAW pulse at the edges of the substrate and due to the electromagnetic crosstalk caused by reflected SAW pulses passing IDT 1. While all these signal components interfere in frequency domain and in this way distort the transmission signal of the main SAW pulse, they can be separated in time domain as they arrive at IDT2 with different delay. To this end, the following steps are applied (cf. [57]): 1. Fourier transformation of the measured S 21 data to time domain. 2. Application of a time gate which excludes all signal components with a delay t < t start or t > t stop. In this way, electromagnetic crosstalk and triple transit are eliminated, as indicated in Fig. 3.7b. 3. Backtransformation of the time-gated data to frequency domain. In Fig. 3.7a, raw and processed frequency domain data are plotted to show the effects of the time gate. While in the raw transmission data, the SAW transmission maximum around ν = 1.55 GHz is superimposed with some oscillating signal, the processed data reveal a clear maximum and the expected sin(ν)/ν -dependence 6. All S 21 data presented in the following are post-processed data, as just explained. In presence of a static external magnetic field, S 21 changes measurably, as can be seen from Fig. 3.8a. Plotting S 21 against the magnetic field strength leads to 6 Theobserved sin(ν)/ν -dependencereflectstheso-calledresponse function oftheidt.itdescribes the frequency-dependent SAW excitation efficiency of the IDT, and can be written as the Fourier transform of the weighting function of the IDT fingers. For a finite number of equal fingers with constant overlap, the weighting function is a rectangle function, whose Fourier transform is given by sin(x)/x, with x = ν/ν 0 and some constant ν 0 [59,60].

52 48 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory -20 a raw data 3 b t start t stop S 21 (db) after timegate S 21 (a.u.) 0-3 EMW SAW triple transit Frequency (MHz) Time t (µs) Figure 3.7: (a) Raw (red) and time-gated (black) transmission data S 21. (b) S 21 as a function of time, obtained by Fourier transformation of the frequency-domain transmission data. EMW is the abbreviation for electromagnetic wave and denotes the electromagnetic crosstalk. S 21 (db) a 150mT =30 7mT Frequency (GHz) S 21 (db) b H sweep direction FMR dips magnetic switching µ 0 H (mt) Figure 3.8: Change of the SAW transmission in a static external magnetic field. In Panel (a), S 21 is plotted against the frequency for two different external fields µ 0 H = 7mT and 150mT. Panel (b) shows the field-dependence of S 21 at the center frequency of the SAW passband, ν = 1.551GHz, at α = 30.

53 Section 3.3 Data processing 49 characteristictransmission curves like Fig.3.8b 7 : Inresonance, i.e. forµ 0 H ±7mT, a strong decrease in transmission of about 5 db can be observed, which corresponds to aneffective damping of about90db/cm caused by ADFMR. This means that FMR can effectively be excited by acoustic phonons in a ferromagnet. Besides the two dips in S 21 around µ 0 H ±7mT, we observe a sharp dip at µ 0 H 2mT. This dip stems from magnetic switching, i.e. the switching of the magnetic moments of the nickel when changing the external magnetic field direction [43], and occurs at µ 0 H = ±µ 0 H coer, where µ 0 H coer = 2mT is the coercive field of nickel [43]; the sign depends on the sweep direction of µ 0 H. As the ADFMR signal due to magnetic switching corresponds to non-resonant interaction between acoustic wave and magnetization, it was not examined further in this thesis. Comparing experiment and calculations, one has to consider that the ADFMR models presented in Secs. 2.1, 2.2 and 2.3 do not include magnetic switching, so that experiment and calculation do not agree at µ 0 H coer. The resonant coupling between SAW and ferromagnet usually occurs at higher fields than µ 0 H coer, thus the comparison of measured and calculated resonant ADFMR signature is not disturbed by magnetic switching SAW power While the transmission coefficient S 21 allows to calculate the damping of the SAW, the reflection coefficient S 11 can be used to determine the power carried by the SAW. For frequencies outside the SAW passband, the output power P 0 of the VNA is partly reflected in IDT 1 and partly emitted in the form of electromagnetic radiation. Within the SAW passband, the reflection coefficient S 11 decreases remarkably due to the energy transport of the SAW, as shown in Fig. 3.9a. With S 11 (EMW) 2 = 0.36 and S 11 (EMW+SAW) 2 = 0.17 from Fig. 3.9a, the power of the SAW is [43] P SAW = P 0 η, (3.3) 2 where we have defined η := S 11 (EMW) 2 S 11 (EMW+SAW) 2. The factor 1/2 in Eq. (3.3) accounts for the fact that there are two SAWs launched by IDT1, propagating in opposite directions [43]. 7 In order to reduce the noise level, Fig. 3.8b does not show the field-dependence of one single frequency point at ν = GHz, but the averaged field-dependence of eight datapoints between 1.547GHz and 1.555GHz. A similar averaging was applied to all data presented in the following.

54 50 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory a S EMW EMW +SAW Frequency (GHz) b P SAW ( W) 25 P 0 =0.1 mw Frequency (GHz) xx 106 Figure 3.9: (a) Square of the reflection coefficient, S 11 2, measured around the 5 th harmonic at 860MHz. η is the power conversion efficiency of the IDT. (b) SAW power and calculated longitudinal strain ε xx for the four SAW passbands at a microwave power of P 0 = 0.1mW. Besides the elastic displacement, also the electrical field induced by the SAW contributes to P SAW. Hereby, the acoustic power of the SAW is P acoustic = κ 2 P SAW [59,61] with the electromechanical coupling factor κ 2 (see Sect. 1.4). Robbins [61] derived an approximative formula which relates the acoustic power of the SAW with the normal component u z of the mechanical displacement: u 2 z = P acoustic 2MπνW, (3.4) with M = Jm 3 for LiNbO 3 [61] and the delayline width 8 W = 500µm. For Rayleigh waves, longitudinal and normal displacement component are related by u x 2 3 u z (cf. Fig. 1.2). Thus, for a plane wave u x exp(ikx), the pure strain ε xx = ux x is ε xx = k u x = 2 3 k P acoustic 2MπνW. (3.5) In Fig. 3.3b, the magnitude of ε xx is plotted for the four SAW passbands. The strains induced by the SAW are in the order of 10 6 corresponding to displacements in the picometer range. The maximum strain is reached at the 5 th harmonic. Because of several approximations in the derivation of Eq. (3.5) and the disregard of 8 Here, we assume the delayline width to be identical with the IDT aperture w IDT, neglecting the divergence of the wave. This is a good approximation, as for the given crystal cuts, the SAW automatically tends to propagate along the destined direction due to the beam steering effect [59].

55 Section 3.3 Data processing 51 the z-dependence of u, Eq. (3.5) is not suitable for an exact calculation of the strains induced by the SAW. Instead, the strains ε ij appearing in the LLG and backaction model were treated as free parameters and fitted to the experimental results. It will be shown below (cf. Tab. 3.3) that the deviation of the present estimate from the fitted strains is less than 30%. This is quite remarkable, considering the fact that Eq. (3.5) is an a priori estimate which uses only the variation in S 11 and the output power of the VNA to predict the SAW displacement Normalized transmission coefficient and absorbed power As one can see from Fig. 3.8, S 21 1 even off resonance, e.g. for µ 0 H = 150mT. The reason is that the microwave provided by the VNA has to be converted into a SAW in IDT1, propagate from IDT1 to IDT2, and be converted back to a AC voltage in IDT2. While the propagation loss of the SAW can be neglected 9, the conversion of a radio-frequency voltage into a SAW and vice versa in the IDTs is comparatively inefficient, leading to a field-independent contribution to the damping of the microwave. As we are only interested in the ADFMR-related SAW damping, which is field-dependent and vanishes for very low/high fields, the measured S 21 is normalized by S norm 21 (µ 0 H) := S 21(µ 0 H) S 21 (µ 0 H off ), (3.6) where µ 0 H off is large compared to the occurring resonance fields. This normalization has been applied to all experimental data shown in the following. Nevertheless, we keep the symbol S 21 instead of S norm 21 for a clearer notation. The power absorbed in ADFMR can be derived from the power carried by the SAW (see Eq. (3.3)) and the normalized transmission coefficient S 21 (see Eq. (3.6)) and is given by P abs = ( 1 S 21 2) P SAW. (3.7) 9 For Rayleigh waves on YZ-LiNbO 3, e.g., the attenuation coefficientis about 1dB/µs at ν = 1GHz [25].

56 52 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory st 5 th 13 th YZ-LiNbO 3 S 21 (db) th Frequency (GHz) Figure 3.10: Transmission spectrum of Sample 1. Besides the SAW frequencies 172 MHz, 860MHz, 1.55GHz and 2.24GHz (the latter are more clear in a zoom-in spectrum), we observe other peaks which are due to resonances in the measurement setup (radio-frequency dipstick, microwave cables, chipcarrier etc.). 3.4 Rayleigh wave driven ADFMR SAW transmission of Sample 1 In Fig. 3.10, the transmission spectrum of Sample 1 is shown. As expected we do not see only the fundamental SAW frequency at 172MHz, but also the 5 th, 9 th and 13 th harmonics. This allows us to study acoustically driven FMR at different frequencies. The measurements presented in the following are confined to a small range of 100MHz around the SAW passband center frequencies 860MHz, 1.55GHz and 2.24 GHz. The fundamental frequency 172 MHz is too low to satisfy the resonance condition for FMR (cf. [18]), so we do not consider this SAW passband further. The measured transmission data evaluated as described in Sect. 3.3 show a distinct magnetic field-dependence, which varies with the angle between external magnetic field µ 0 H and the SAW propagation direction x. Therefore, in the following sections, magnitude and phase of S 21 are plotted against magnetic field strength and orientation in order to show the full characteristic signature of ADFMR.

57 Section 3.4 Rayleigh wave driven ADFMR Comparison of the experimental data with the ADFMR models Before the experimental results are shown for all investigated frequencies and all magnetic field orientations, we first present a comparison of one exemplary ADFMR trace with the modelling approaches presented in Chap. 2. To this end, in Fig. 3.11, S 21 as a function of magnetic field strength and orientation is plotted for ν = 1.55GHz and IP configuration, together with the simulation results obtained with the LLG, backaction and SAW model, respectively. ψ denotes the angle between µ 0 H and the SAW propagation direction x, as indicated in Fig For a better understanding of the color code in Fig. 3.11, we refer to Fig. 3.8b, which shows a single ADFMR trace (at α = 30 ) out of Fig. 3.11b. The simulations requireaset of parameters{b d,b u,a}, which hasto bedetermined from the comparison of experimental and calculated data. We found that B d = 400mT and B u = 2.5mT along u = x allow to consistently describe the experimental results (see Figs to 3.14). The damping parameter was found to be a = 0.1, which is a factor of two larger than typical literature values for polycrystalline nickel films [62]. The reason for the larger a-value could be the excitation of higher spin wave modes 10 which cannot be resolved, or non-gilbert-type damping contributions. Besides, surface pinning of the electron spins affects the magnetization excitation and has been reported to broaden the FMR line[49,63]. This effect increases with the wave number k and therefore could make a relevant contribution in ADFMR, where we are dealing with comparatively short wavelengths and high wave numbers. Therefore, further studies of ADFMR at higher frequencies are required. The material parameters for the nickel film have been taken from Tab The magnetoelasticcouplingconstantb 1,usedinthebackactionmodel,hasbeencalculated via b 1 = 3λ s µ Ni = 23T [46] with the isotropic magnetostriction constant λ s and the Lamé constant µ Ni from Tab The thickness of the nickel film in Sample 1 is t Ni = 50nm (see Tab. 3.1). The strains fitted with the LLG and backaction model and the filling factor used in the backaction model are given in Tab The material constants for LiNbO 3, which have been used in the SAW model, are given in Tab The effective magnetoelastic coupling constant for the SAW model was chosen b 1 = 14T. 10 In the presented models, we assumed a uniform magnetization with all electron spins precessing in phase. This corresponds to the fundamental (collective) spin wave mode.

58 54 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory Figure 3.11: Comparison between measured and simulated angle-dependence of ADFMR in Sample 1, using the models presented in Chap. 2. Panel (a) to (c) show P abs, S 21 andarg(s 21 ) plotted versusmagnetic fieldorientation andstrength for IP configuration and ν = 1.55GHz. In Panel (d), the power absorption calculated with the LLG approach (see Eq. (2.12)) is shown. In Panel (e) and (f), S 21 and arg(s 21 ) are plotted as calculated using the backaction model (see Eq. (2.25)). Panel (g) and(g), finally, show S 21 andarg(s 21 )calculated with the SAW model. The parameters used for the simulations are listed in the text and in Tab As magnitude and phase of S 21 already contain the full information about ADFMR, we have renounced on the additional calculation of the power absorption with the SAW model.

59 Section 3.4 Rayleigh wave driven ADFMR 55 As one can see from Fig. 3.11, the main features of the experimentally determined ADFMR signature are reproduced in the models 11. Besides, the results obtained by the different ADFMR models are very similar, demonstrating that the presented modelling approaches are principally consistent. Therefore, for the detailed discussion of Rayleigh wave driven FMR in the rest of this section, we use the LLG and backaction model to reproduce the measured data. We renounce on showing the results from the more complex SAW model as they would give no additional insights into the physics of Rayleigh wave driven FMR. For the comparison of shear and Rayleigh wave driven FMR (see Sect. 3.5), we use the SAW model as it inherently includes the different strains induced by SH and Rayleigh wave. Accordingly, the strains are not treated as fit parameters (as it is the case for the LLG and backaction model) but are directly determined in the SAW model calculations Angle-dependent ADFMR In this section, the complete set of measurement and simulation data for the Rayleigh wavedrivenfmrexperiments withsample 1isshown. InFig.3.12,themeasuredand simulated angle-dependence of ADFMR in IP configuration (cf. Fig. 3.5) is plotted for the 5 th, 9 th and 13 th harmonic frequency, corresponding to ν = 860MHz,1.55GHz and 2.24 GHz. The experimental data were obtained using the microwave power P 0 = 10dBm and the IF bandwidth of 2kHz (at 860MHz and 1.55GHz) resp. 1kHz (at 2.24GHz). As expected [48,49], the resonance shifts to higher fields µ 0 H with increasing frequency, as can be seen in Fig For small frequencies, the quantitative agreement between simulation and experiment is very good, whereas for higher frequencies, P abs and S 21 are overestimated in the simulations. The reason is that the fit parameters were chosen such that the phase quantitatively agrees with the experiment. The quantitative comparison of the magnitude then shows some deviations, but we observe a quite reasonable agreement as well. Besides, we notice a slight asymmetry in the experimental data, which goes back to the contribution of the shear strain ε xz and can be reproduced in the LLG model (see Sect ). For the backaction model, however, it is hard to consider a shear strain 11 As explained in Sect , the magnetic switching at about µ 0 H = 2mT is non-resonant and not accounted for in the ADFMR models.

60 56 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory 0.86GHz 1.55GHz 2.24GHz ε xx (10 6 ) ε xz (10 6 ) 0.09i i 0.018i ε zz (10 6 ) F (d/λ) Table 3.3: Frequency-dependent fit parameters used for the simulations in Fig. 3.11, 3.12, 3.13 and in addition to the dominant longitudinal strain ε xx. Therefore, ε xz was assumed to be zero yielding symmetric ADFMR signatures. Regarding the phase of S 21, we experimentally observe phase shifts up to 115. As explained in Sect. 2.2, the backaction model, as presented here, only holds in the limit of small interaction between SAW and FM, which implies a transmission S 21 near 1 and a phase shift which is small compared to 90. Thus, the observed strong interaction between SAW and FM is beyond the validity of the backaction model. In order to justify the disregard of the ferromagnetic exchange interaction in the ADFMR models (see Sect ), we calculate its contribution µ 0 H ex m to the freeenthalpy density G, which is given by Eq. (2.2). With µ 0 H ex = D s m [19,45] and D s = Tm 2 [64], we get a resonance shift of only 0.4mT even at the highest examined SAW frequency ν = 2.24GHz. This is small compared to the other contributing terms in Eq. (2.2) and therefore does not change the ADFMR signature significantly. In Fig and 3.14, the experimental and simulated ADFMR data are plotted for theoop1 and OOP2 configuration. These dataareonekey result of thisthesis, asto our knowledge no such SAW driven ADFMR experiments have been published before. For the calculation, the same set of parameters was used as for the IP configuration. Instead of the butterfly shape observed in IP configuration, we now see a cross-like signature, with an approximately four-fold symmetry as well. The small asymmetry with respect to the angle ψ is due to a slight misalignment of the sample, which was about1.7 in OOP1 configuration and0.6 inoop2 configuration 12, andwhich could 12 This means that the real rotation plane of the magnetic field was not exactly perpendicular but slightly tilted towards the film plane, which is due to an imprecise mounting of the sample in the rf dipstick. The misalignment could not be measured directly but was determined by fitting the rotation plane in the simulation to the measured ADFMR data.

61 Section 3.4 Rayleigh wave driven ADFMR 57 Figure 3.12: Measured and simulated angular-dependence of ADFMR in IP configuration (cf. Fig. 3.5) in Sample 1, for the SAW frequencies 860MHz, 1.55GHz and 2.24GHz. The three columns show P abs, S 21 and arg(s 21 ) plotted versus magnetic field orientation and strength. The simulated power absorption was calculated using the LLG approach (see Eq. (2.12)), magnitude and phase of S 21 were obtained from the backaction model (see Eq. (2.25)). The used parameters are listed in the text and in Tab. 3.3.

62 58 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory be reproduced in the simulation very well. As in IP configuration, we observe a sign change of arg(s 21 ) for higher frequencies, which is also reflected by the calculation results. Comparing the strain estimated from S 11 (see Fig. 3.9) with the fitted longitudinal strain listed in Tab. 3.3 shows a good agreement with deviations of 30% or less and confirms the choice of the fit parameters ε ij a posteriori. In conclusion, we observe an excellent agreement of the experimental data with the calculation results. Except from the damping parameter a, which has been discussed above, all simulation parameters have been chosen consistent with literature. Moreover, all investigated magnetic field configurations (IP, OOP 1 and OOP 2) were calculated using one set of parameters, which demonstrates the reliability of the presented ADFMR models. 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR SAW transmission of Sample 2 Besides a detailed study of Rayleigh wave driven FMR on YZ-LiNbO 3, in this thesis the effects of the SAW type on the ADFMR signature were investigated. As already mentioned in Chap. 1, 36 YX-LiTaO 3 supports SH waves as well as Rayleigh waves, but the coupling factor κ 2 for the Rayleigh wave is about two orders of magnitude smaller than for the SH wave. Therefore, the Rayleigh wave mode is usually neglected in SAW applications. Nevertheless, by adjusting microwave power, IF bandwidth and other measurement parameters, it was possible to reduce the noise level sufficiently to get a clean ADFMR signal not only for the SH wave, but also for the Rayleigh wave in Sample 2. Thereby, for the latter, the transmission at 780MHz (the 5 th harmonic of the Rayleigh fundamental frequency) was not more than 60 db. Figure 3.15 shows an overview of the transmission spectrum of Sample 2 (36 YX- LiTaO 3 with a 50nm thin Ni film) with the frequencies indicated at which either Rayleigh or shear horizontal waves are excited. In Fig. 3.16, an enlarged view of the transmission spectrum around the 5 th harmonic of the Rayleigh wave and the 5 th and 9 th harmonic of the SH wave is shown. One can see that the Rayleigh wave is very

63 Section 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR 59 Figure 3.13: Measured and simulated angular-dependence of ADFMR in OOP 1 configuration (cf. Fig. 3.5) in Sample 1, for the SAW frequencies 860MHz, 1.55GHz and 2.24GHz. The three columns show P abs, S 21 and arg(s 21 ) plotted versus magnetic field orientation and strength. The simulated power absorption was calculated using the LLG approach(see Eq.(2.12)), magnitude and phase of S 21 were obtained from the backaction model (see Eq. (2.25)). The used parameters are listed in the text and in Tab. 3.3.

64 60 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory Figure 3.14: Measured and simulated angular-dependence of ADFMR in OOP 2 configuration (cf. Fig. 3.5) in Sample 1, for the SAW frequencies 860MHz, 1.55GHz and 2.24GHz. The three columns show P abs, S 21 and arg(s 21 ) plotted versus magnetic field orientation and strength. The simulated power absorption was calculated using the LLG approach(see Eq.(2.12)), magnitude and phase of S 21 were obtained from the backaction model (see Eq. (2.25)). The used parameters are listed in the text and in Tab. 3.3.

65 Section 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR 61 S 21 (db) st R 1 st SH 5 th R 5 th SH 9 th SH YX-LiTaO Frequency (GHz) Figure 3.15: Transmission spectrum of Sample 2. The fundamental and the 5 th harmonic of the Rayleigh wave (at 156MHz and 780MHz) are indicated, as well as the fundamental, 5 th and 9 th harmonic of the shear-horizontal wave (at 206MHz, 1.04GHz and1.87ghz). Similarto Fig. 3.10, thereareother peaks stemming from resonances in the measurement setup. weak compared to the SH wave at 1.04GHz 13. Nevertheless, Fourier transformation and time-gating, as described in Sect , reveals a transmission maximum at about 780 MHz. In order to demonstrate that we really observe two different surface wave types on one and the same sample, the transmission S 21 for the passbands around 780MHz, 1.04GHz and 1.87GHz is plotted as a function of time in Fig From the timedomain transmission, we can read off the delay t of the respective wave and in this way derive its speed of sound c sound = d IDT /t. The obtained values for t and c sound are listed in Tab Comparing the sound velocities with literature data(e.g.[40]) shows that the wave excited at 780MHz is Rayleigh-type, whereas the SAWs at 1.04GHz and 1.87GHz are SH waves Angle-dependent ADFMR For a comparison between Rayleigh and SH wave driven FMR, the 5 th harmonics of both wave types at 780MHz resp. 1.04GHz were studied in IP and OOP1 configura- 13 The 9 th harmonic of the SH wave is very weak, too, but this is due to the fact that the SAW excitation efficiency of an IDT decreases for higher harmonics.

66 62 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory S 21 (db) a after timegate raw data b after timegate raw data Frequency (GHz) c raw data after timegate Figure 3.16: Magnitude of the SAW transmission of Sample 2 as a function of the microwave frequency ν for (a) the Rayleigh-wave passband around 780 MHz, (b) the SH-wave passband around 1.04 GHz and (c) the SH-wave passband around 1.87GHz. The red and black lines show the raw and the time-gated transmission data, respectively. 0.5 t 1.04GHz t 780MHz S 21 (a.u.) MHz passband (Rayleigh) t 1.87GHz 1.04GHz passband (SH1) 1.87GHz passband (SH2) Time (µs) Figure 3.17: S 21 as a function of time (Sample 2). The black, red and green line denote the SAW passbands around 780 MHz, 1.04 GHz and 1.87 GHz, respectively. This plot reveals different delay times of the examined SAWs and in this way allows to determine the type of the excited waves.

67 Section 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR 63 frequency ν delay t c sound wave type GHz µs m/s Rayleigh Shear horizontal Shear horizontal Table 3.4: SAW delay t, determined from Fig. 3.17, and calculated sound velocity c sound = d IDT /t, with d IDT = 5.8mm (cf. Tab. 3.1). The wave type was found by comparing c sound with the literature data in Tab th R 5 th SH 9 th SH frequency ν GHz wavelength λ 20/5 20/5 20/9 µm VNA power P dbm IF bandwidth khz Table 3.5: Measurement parameters used for the Rayleigh and SH wave driven FMR experiments on Sample 2. tion. For the IP configuration, also the 9 th harmonic of the SH wave at 1.87GHz was investigated. The measurement parameters are listed in Tab Figure 3.18 shows the experimentally determined angle-dependence of ADFMR in IP configuration, driven by Rayleigh and shear-horizontal wave and measured in Sample 2. It confirms the calculation results from Sect regarding the different angle-dependence of Rayleigh and SH wave driven FMR: While the Rayleigh wave FMR signature (first column in Fig. 3.18) resembles the ADFMR measurements on LiNbO 3 14 (cf. Figs. 3.12), SH wave FMR shows maximum signal at 0 and ±90 and a minimum around ±50. This can be understood regarding the different angledependence of the virtual driving field µ 0 h, depending on the wave type, as discussed in Sect Besides, SH wave driven FMR shows an astonishing behaviour regarding two as- 14 As already mentioned in Sect , the Rayleigh wave on 36 YX-LiTaO 3 is not completely equal to the oneonyz-linbo 3, sowedonotexpectaperfectagreement. In particular,onyz-linbo 3, the only non-vanishing shear strain component is ε xz, whereas on 36 YX-LiTaO 3, ε xz = 0 with non-vanishing ε yz and ε xy. Therefore, the asymmetry in the ADFMR signature, which stems from the shear strains, looks different for the two crystals.

68 64 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory Figure 3.18: Measured angle-dependence of ADFMR in Sample 2 in IP configuration for Rayleigh wave (at 780MHz) and SH wave (at 1.04GHz and 1.87GHz). In the first row, S 21 is plotted versus magnetic field orientation and strength, the second row shows arg(s 21 ). pects: First, we observe areas in Fig. 3.18, where the transmission magnitude S 21 clearly exceeds one. This means that the SAW propagation is effectively enhanced by interaction with the ferromagnetic film. The second aspect is that the 9 th harmonic of the SH wave (at 1.87GHz) shows to some extent the opposite behaviour of the 5 th harmonic. Not only the sign of the phase shift is reverse (positive at 1.04GHz, negative at 1.87GHz; see also Fig. 3.19), also the broad resonance peaks at ψ = 0 and ±90 are directed towards lower and higher transmission, respectively. Both observations do not appear in Rayleigh wave driven ADFMR and cannot be modelled yet. Nevertheless, we want to give some possible explanation approaches which could

69 Section 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR 65 Figure 3.19: Shift of transmission, S 21 := S 21 S 21 (µ 0 H off ), with respect to the external magnetic field, as a function of frequency (Sample 2). Panels (a) and (b) show the 5 th and the 9 th harmonic of the SH wave, respectively. µ 0 H off was taken as 150mT for the 5 th harmonic and 250mT for the 9 th harmonic. As one can see, the field- and frequency-dependent phase shift looks very similar for 5 th and 9 th harmonic of the SH wave, except for the different sign. account for these phenomena: The observation of a normalized transmission coefficient S 21 > 1 suggests that there are other, non-resonant damping mechanisms in addition to the resonant damping due to ADFMR. It seems plausible that the AC shear strains in the nickel film lead to a considerable mechanical damping, which occurs independently of FMR. This is not accounted for in the ADFMR models as we assume an ideal harmonic coupling between the nickel atoms without any dissipative terms. It is known [65 67] that in FMR the elastic constants of the FM can change; this phenomenon is called Eeffect [65]. According to [68], the relative variation of the elastic modulus of nickel can be up to 18% depending on the magnetization state. We thus speculate that the change in the material constants is such that it counteracts the dissipative terms and in this way reduces the effective damping of the acoustic wave in FMR. As the experimental data are normalized such that off resonance S 21 = 1, this yields transmission values greater than 1 in FMR. Another issue which has not been completely understood yet is the excitation and detection of shear waves using IDTs. For a Rayleigh wave the response of the piezoelectriccrystaltotheacelectricfieldoftheidtcaneasilybeexplained(seee.g.[43]), as the piezoelectric crystal is compressed and strained in the direction of the electric

70 66 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory Figure 3.20: Measured angle-dependence of ADFMR in OOP 1 configuration for Rayleigh wave (at 780MHz) and SH wave (at 1.04GHz) in Sample 2. In the first row, S 21 is plotted versus magnetic field orientation and strength, the second row shows arg(s 21 ). field. For shear waves, however, the dominant displacement is perpendicular to the electric field direction. Hence, coupling between shear wave and electrical field is presumably mediated by a longitudinal/normal acoustic wave, which, however, decays exponentially along the surface. The surprising behaviour of the phase shift (see Figs and 3.19) is possibly due to such a rather complicated interaction mechanism of IDT and shear wave. To clarify this issue would require a further experimental and theoretical study of the working principle of IDTs for shear waves. Concerning the OOP 1 configuration, which is plotted in Fig. 3.20, the difference between Rayleigh and SH wave FMR is less distinct. In both cases, we observe

71 Section 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR 67 a cross-like signature with its center at ψ = 0 (corresponding to H k SAW ) and µ 0 H = 0. Nevertheless, for a fixed angle ψ, the resonance fields of SH wave driven FMR are much higher than those of Rayleigh wave FMR. This again reflects the different angle-dependence of the virtual driving field for Rayleigh and SH wave. The phase (second row in Fig. 3.20), shows similar angle-dependence as S 21, where the phase shift is negative for the Rayleigh wave and positive for the SH wave, as already observed in IP configuration Comparison of the experimental data to the SAW Model Having shown the measured angle-dependence of Rayleigh and SH wave driven FMR insample 2,weusetheSAWmodeltosimulatetheexperimentaldata. Thereby, wedo not have to make any assumptions concerning the strains induced by the different wave types, as it was necessary in the backaction model. The SAW model rather includes the calculation of the wave type based on the material constants of the crystal and the applied microwave frequency. Apart from the anisotropy parameters B u and B d, we thus have only two free parameters, the effective magnetoelastic coupling constant b1 and the damping parameter a, which have to be fitted to the experimental data. Hereby, B u, B d and b 1 reflect the material system and the geometry of the sample and are therefore independent of wave type and frequency. For the anisotropy parameters, we use B u = 0.4mT along u = x and B d = 350mT, the effective magnetoelastic coupling constant was chosen b 1 = 8T and the damping parameter a was 0.16 (for the Rayleigh wave at 780MHz) resp. 0.5 (for the SH wave at 1.04GHz). This set of parameters turned out to yield the best agreement with the experimental data. Figures 3.21 and 3.22 show the measured and calculated transmission S 21 for the 5 th harmonic of Rayleigh and SH wave in IP and OOP1 configuration. As one can see, the basic angle-dependence of Rayleigh and SH wave driven FMR can be reproduced in the simulations with one and the same set of parameters, but in detail there are still differences. Nevertheless, the experimentally determined and the calculated transmission S 21 differ by a factor of 1.5 at most, which makes it possible to use the same color code for both experiment and simulation in Figs and Besides, for the simulation of IP and OOP1 configuration, the same set of parameters was used, demonstrating the consistency of the model with the experimental data. Concerning the deviations between experiment and simulation, there can be several

72 68 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory Figure 3.21: Comparison of measured and calculated ADFMR signature in IP configuration for Rayleigh wave (at 780MHz) and SH wave (at 1.04GHz) in Sample 2. The simulation results were obtained from the surface acoustic wave model as explained in Sect The parameters used for the calculation are given in the text.

73 Section 3.5 Comparison of shear-horizontal and Rayleigh wave ADFMR 69 Figure 3.22: Comparison of measured and calculated ADFMR signature in OOP 1 configuration for Rayleigh wave (at 780MHz) and SH wave (at 1.04GHz) in Sample 2. The simulation results were obtained from the surface acoustic wave model as explained in Sect The parameters used for the calculation are given in the text.

74 70 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory reasons: First, the material parameters found in literature are not fully consistent, but show differences up to 10%. This affects the ADFMR calculation results as the displacements and strains induced by the SAW are determined by the exact material properties. Moreover, the crystal cut of the used LiTaO 3 substrate is given only with an accuracy of 1 and does not exactly match the ideal value found in literature, which is 34.7 for a metallized surface and 35.1 for a free surface. Also the alignment of the crystal edges and the IDTs regarding the crystal X-axis presumably comprises an error of some degrees. Finally, the conversion of an AC voltage into a SAW and vice versa in the IDTs brings about further uncertainties which are not accounted for in the ADFMR models. 3.6 Summary In this chapter, the experimental ADFMR data have been presented and compared to the experiment. First, an extensive study of Rayleigh wave driven FMR on YZ- LiNbO 3 was given: We measured angle- and field-dependent ADFMR for three different frequencies between 860 MHz and 2.24 GHz and three measurement configurations (IP, OOP 1 and OOP 2), and showed that the experimental data can quantitatively be understood on the basis of the three ADFMR models introduced in Chap. 2. Considering that the original modelling and theory considerations by M. Weiler [16] were based on in-plane experiments only, the presented OOP 1 and OOP 2 measurements, studied for the first time in this thesis, are an important step forward, corrobating the current understanding of magnetoelastic interaction and ADFMR. Furthermore, the symmetry break of the characteristic four-fold ADFMR signature, which was derived theoretically in Chap. 2, indeed could be observed in the experiment. Moreover, it has been shown, that the three ADFMR models yield very similar results and are therefore consistent in the limit of small interaction between SAW and FM. Nevertheless, they are different concerning range of validity, modelling sophistication and numerical complexity. Besides, the effects of the SAW type on the ADFMR signature have been studied. To this end, a sample was fabricated which allows to measure Rayleigh and SH wave driven FMR at the same time in the same sample. We would like to stress that shear wave driven FMR has been reported in this thesis the first time. The fundamental differences i. e. the observed different angle-dependence and the occurring symme-

75 Section 3.6 Summary 71 tries could be understood on the basis of analytical considerations (see Sect ) as well as numerical calculations. In detail, however, there are differences between experiment and simulation which have not been understood yet and therefore require further investigation.

76 72 Chapter 3 Acoustically driven FMR: Experimental study and comparison to theory

77 Chapter 4 Tuneable surface acoustic wave resonator 4.1 SAW resonators and overview All the ADFMR experiments presented in Chap. 3 have been performed on SAW delaylines. In other words, each sample consists of a pair of IDTs, which are used to launch and detect the SAW, respectively. As shown in Sect , the SAW is partly reflected at the IDTs, which results in the observation of a triple transit SAW pulse (cf. Fig. 3.7b). Nevertheless, the reflexion coefficient of an IDT is comparatively small so that the SAW between IDT1 and IDT2 is mainly a travelling wave with propagation speed v m/s. The amplitude of the SAW is determined by the microwave power P 0 input into IDT1, the SAW excitation efficiency of the IDT (given by η = S 11 2, cf. Sect ) and the coupling coefficient κ 2 between elastic and electrical component of the SAW (cf. Sect. 1.4). Hereby, S 11 2 and κ 2 depend on the material system and cannot be increased directly, and P 0 is limited by the maximum current density supported by the IDT stripes. In order to exceed these limitations and further increase the SAW amplitude one needs to go beyond the simple delayline concept. One attractive alternative are SAW resonators. ADFMR experiments using SAW resonators instead of conventional delaylines would entail several advantages: First, due to the increase in mechanical displacement and induced strains, the ADFMR signal could be raised, yielding a better signal-to-noise ratio and simplifying measurement and evaluation techniques. Besides, very small ADFMR-related signals like acoustic spin pumping DC as well as AC could better be resolved and studied 73

78 resonator given by 1 2d = Nλ, N N, (4.1) 74 Chapter 4 Tuneable surface acoustic wave resonator in more detail. Second, the increase in SAW amplitude could allow to exceed the range of small magnetization excitation m 1,2 1 in order to observe and examine nonlinear effects in ADFMR and acoustic spin pumping. Third, in a resonator, the reflected wave superimposes the wave launched by IDT 1, which results in a standing wave. It would be interesting to perform ADFMR experiments with standing waves instead of travelling waves, as this allows for spatially confined excitation and thus for spatially resolved ADFMR and spin pumping. For ADFMR applications, it is essential to be able to tune the SAW resonator, which is explained in the following: The condition for constructive interference of incoming and reflected wave (resonance condition) is independent of the type of where λ is the wavelength of the resonator wave and d the (effective) distance between the reflectors [25]. An additional load in the resonator changes the propagation speed of the wave and thus the wavelength λ (for fixed frequency). Such a load can be e. g. an additional capacity in an electromagnetic oscillating circuit, a magnetic solid in a microwave cavity or a ferromagnetic film on a SAW resonator. In particular, this implies that a SAW resonator well tuned in its empty configuration will not be appropriately tuned if a ferromagnetic thin film is evaporated onto it. Thus, the resonance condition Eq. (4.1) may not be satisfied any more in SAW resonators loaded with a FM. Therefore, we need a possibility to tune the resonator in-situ, i.e. to vary its acoustic length slightly, in order to satisfy Eq. (4.1). Depending on the specific type of resonator, there are different ways of tuning. For electromagnetic oscillating circuits, e. g., this can be done by changing the capacity/inductivity of the circuit, and in a laser cavity, the distance between the mirrors can be adjusted. In a SAW resonator, however, the reflectors consist of lithographically defined gratings on the crystal surface and therefore the distance between the reflectors is fixed. A possibility to tune a SAW resonator would be to vary outer parameters like the temperature Θ, as the stiffness constants of the crystal and therefore also the sound velocity v changes with temperature. An estimation of the needed Θ is given in the outlook (Sect. 5.2). 1 Here, we have assumed that the SAW reflectorscause a phase shift φ refl {0,π} each so that the total phaseshift ofthe SAWpulse atboth ends ofthe resonatoris φ refl,ges = φ refl,1 + φ refl,2 2πN.

79 Section 4.2 Flipchip delayline 75 The disavantage of tuning via temperature is that nearly all relevant material constants show a certain temperature dependence and so not only the speed of sound of the SAW but also the elastic and magnetoelastic properties of the ferromagnetic film change. Therefore it is desirable to work at constant temperature and vary the distance between the reflectors without changing any other material property, as realized in the present work. Within the scope of this diploma thesis, it has been shown that a SAW resonator can be designed such that the reflectors are patterned on two independent crystals which are then assembled on a chipcarrier with an air gap of some millimeters in between, as indicated in Fig One of the crystals is fixed to the chipcarrier, while the other one can be moved with a piezoelectric actuator by some microns. To enable SAW propagation from one crystal to the other (across the air gap), a third, blank bridge crystal is added on top (see Fig. 4.1b). In the following, we will show that this design of a mechanically tuneable SAW resonator works and allows to tune the transmission S 21 over a wide range. The outline of this chapter is as follows: First, we show that a SAW can effectively couple from one ferroelectric crystal into another which lies on top of the first. Then, a tuneable delayline is presented which allows the direct measurement of the distance variation via vector network analysis. Last, the tuneable SAW resonator is characterized. We observe strong variations in transmission amplitude and phase dependent on the voltage applied to the piezoelectric actuator. The samples and experiments presented in this chapter have been made together with my bachelor student Alexander Späh. In his bachelor s thesis [69], the complete set of experimental data is shown whereas here only the most important results are presented. 4.2 Flipchip delayline As a first step, we demonstrate that SAWs effectively couple from one ferroelectric crystal to another which is on top of the first as illustrated in Fig To this

80 76 Chapter 4 Tuneable surface acoustic wave resonator a b IDT IDT IDT bridge IDT LiNbO 3 LiNbO 3 LiNbO 3 Figure 4.1: Schematic view of (a) a conventional delayline and (b) a flipchip delayline. end, a delayline with two IDTs was fabricated on a YZ-LiNbO 3 crystal, which was subsequently cut in two equal pieces 2. The crystal halves were aligned parallel and fixed on a chipcarrier, such that an air gap of about 1.1mm separated them. Then another Y-cut LiNbO 3 crystal with dimensions 5 7mm was set on top of the air gap with the polished side downwards and its Z-axis aligned parallel to the Z-axes of the crystal halves. The overlap of this bridge crystal with the delayline halves was approximately 2 mm on each side. We refer to this unconventional delayline as the flipchip delayline in the following. The flipchip delayline was characterized by measuring the complex transmission coefficient S 21, and compared to a standard delayline with the same fabrication parameters and d IDT = 5.8mm. Electromagnetic crosstalk, triple transit and other spurious signal components were eliminated by application of a time gate, as described in Sect In Fig. 4.2, S 21 2 is plotted for both flipchip and conventional delayline at the fundamental SAW frequency 172MHz and its 5 th harmonic. Compared to the reference delayline, the transmission of the flipchip delayline is lower by about 2dB and 4dB at the fundamental frequency and its 5 th harmonic, respectively. Considering, that the SAW has to couple twice between two crystal surfaces lying on top of each other, the obtained transmission is remarkably high. In conclusion, we see that the flipchip design works and that the loss of transmission compared to a conventional delayline is surprisingly low. This lays the foundation for further experiments with flipchip delayline and flipchip resonator. 2 For the cutting process, we covered the crystal with a thick layer of photoresist to avoid damage to the polished surface.

81 Section 4.3 Tuneable delayline 77 S 21 (db) a flipchip conventional Frequency (MHz) b conventional flipchip Frequency (MHz) S 21 (db) Figure 4.2: Magnitude S 21 2 of the complex transmission coefficient as a function of the microwave frequency for the flipchip delayline (red) and a conventional reference delayline (black). Here, the SAW passbands around (a) 172 MHz and (b) 860MHz are shown. 4.3 Tuneable delayline In order to show that the separation of the IDTs (resp. the air gap length) can be detected by vector network analysis, a tuneable delayline was fabricated (see Fig. 4.3). First, a pair of IDTs was patterned on a 6 9mm 2 YZ-LiNbO 3 crystal, using the same IDT geometry and fabrication parameters as for the flipchip delayline. As above, the delayline was cut in two equal pieces. One of the crystal halves was fixed to the chipcarrier using the two-component adhesive UHU plus sofortfest 2-K- Epoxidharzkleber, the other piece was glued with its end face to a brass block fixed to one side of a piezoelectric actuator (PSt 150/2x3/5 by Piezomechanik München GmbH, see [70]). The other side of the actuator was fixed to the chipcarrier using a second brass block. The bridge crystal was put on top of the 2.4mm wide air gap, with an overlap of approximately 1.3 mm to the crystal halves, and pressed downwards using a brass clip. The pressure can be adjusted at two screws which connect the clip to the chipcarrier. The voltage V piezo applied to the piezoelectric actuator is provided by a Keithley 2400 SourceMeter. As in Sect. 4.2, the complex transmission S 21 of the delayline was measured within the SAW passbands around 172MHz and 860MHz. The magnitude and the phase of S 21 (after time gate) is plotted against the microwave frequency ν for three different piezovoltages in Figs. 4.4 and 4.5. The variation of S 21 with V piezo is rather small, as

82 78 Chapter 4 Tuneable surface acoustic wave resonator piezoel. actuator flexible chipcarrier bridge fixed Figure 4.3: Schematic view of the tuneable delayline. For a photograph, we refer to Fig. 4.9, where the tuneable resonator, which is assembled similar to the tuneable delayline, is shown. expected, whilearg(s 21 )canbeadjustedby50 (at172mhz)resp. 240 (at860mhz). To compare this to the stroke of the piezoelectric actuator, we consider that the change l inthedelaylinelengthisproportionaltothephaseshift ϕofthetransmission coefficient S 21, and that a variationof l by a full wavelength, l = λ, corresponds to ϕ = 360. Therefore, we have l λ = ϕ 360. (4.2) According to its specifications [70], the expected stroke of the piezoelectric actuator is 3 l max := l(v piezo = 100V) l(v piezo = 30V) 4.5µm. Using Eq. (4.2), this corresponds to ϕ 1 = 80 and ϕ 5 = 400 for the fundamental frequency and the 5 th harmonic, respectively. Experimentally, we thus only observe about 60% of the phase change expected from the simple model. The deviation between the experimentally determined phase shift and the calculated ϕ can easily be rationalized as under mechanical load the (some ten microns thick) glued joints between crystal half, brass block, piezoelectric actuator and chipcarrier are deformed elastically by some percent of their thickness. Therefore, the real displacement of the flexible crystal half is less than the stroke l of the piezoelectric actuator. Besides, the translation of IDT 2 can be observed directly using an optical microscope, as shown in [69]. The results obtained optically confirm the specifications of 3 Strictly speaking, the specification of the maximum stroke l max, as given in [70], is valid only for vanishing load. Nevertheless, the maximum load force of the piezoelectric actuator is much higher than the estimated force applied in our case, so the stroke will be approximately the same as without load.

83 Section 4.3 Tuneable delayline 79 S 21 (db) a -30V b -30V 0V 0V 100V 100V S 21 (db) Frequency (MHz) Frequency (MHz) Figure 4.4: Frequency-dependence of the transmission magnitude S 21 for three different piezovoltages V piezo = 30,0,100V around (a) 172MHz and (b) 860MHz. Phase shift ( ) a 100 V 0 V =50 b 100 V 0 V = Phase shift ( ) Frequency (MHz) Frequency (MHz) 0 Figure 4.5: Frequency-dependenceofthetransmissionphaseshift ϕ := arg(s 21 )(V piezo ) arg(s 21 )( 30V) for piezovoltages V piezo = 0V and 100V around (a) 172MHz and (b) 860MHz.

84 80 Chapter 4 Tuneable surface acoustic wave resonator a 172 MHz b 860 MHz S 21 (db) S 21 (db) V piezo (V) V piezo (V) Figure 4.6: S 21 2 as a function of the piezovoltage V piezo at (a) 172MHz and (b) 860MHz. the piezoelectric actuator mentioned above, but cannot be directly compared to the measured ϕ as they have been recorded without load. Nevertheless, it shows that the moving mechanism of the ferroelectric crystal works and that the maximum shift is of the expected order. In order to examine the variation of S 21 with V piezo in more detail, we choose the center frequencies of the SAW passbands (172 MHz and 860 MHz, respectively) and plot S 21 andarg(s 21 )againsttheappliedpiezovoltagev piezo, asshowninfigs.4.6and 4.7. Hereby, V piezo was swept up and down between 30V and 100V several times. Figure 4.6 confirms the observation from Fig. 4.4 that S 21 only slightly depends on V piezo. In contrast, the phase, plotted in Fig. 4.7, varies continuously over a wide range and shows a pronounced hysteresis loop, which results from the characteristic hysteretic behaviour of the piezoelectric actuator. This will be discussed later in more detail (see Sect ). Figure 4.7 demonstrates that a variation in the length of the delayline can quantitatively be detected by analysing the phase of S Tuneable resonator Having shown the functionality of the tuneable delayline, we now turn to the tuneable SAW resonator. The design of the tuneable resonator, sketched in Fig. 4.8, is the same as for the tuneable delayline except for the lithography structure fabricated on the piezoelectric crystal halves. Here, we used IDTs with N = 20 finger pairs and a

85 Section 4.4 Tuneable resonator 81 Phase shift ( ) a up down MHz V piezo (V) b up down MHz V piezo (V) Phase shift ( ) Figure 4.7: arg(s 21 ) as a function of the piezovoltage V piezo at (a) 172MHz and (b) 860MHz. The red and black lines refer to up- and downsweep, respectively. periodicity p = 20µm and an additional reflective grating consisting of 5µm wide aluminum stripes with a periodicity of p = 10µm. The resonator design is that of a conventional two-port resonator [23]. This means that there are two IDTs for launching and detecting the SAW, respectively, located on the delayline in-between the reflective gratings 4. Grating and IDT were fabricated in the same lithographical process. The distance between both is such that the fingers of the IDT are located exactly on the antinodes of the standing wave in the resonator, as explained, e.g., in [60, 69]. The grating stripes are shorted. Similar to the tuneable delayline, one of the crystal halves is fixed to the chipcarrier, the other half can be moved with a piezoelectric actuator (model PSt150/2x3/5), as can be seen from the photograph in Fig The air gap is about 1.9mm. A brass clip presses the bridge crystal onto the resonator halves. On the chipcarrier, there are two mini-smp connectors to which the IDTs are bonded. Using coaxial microwave cables, the sample is connected to a two-port VNA (Rohde&Schwarz ZVB8). As above, the voltage applied to the piezoelectric actuator is provided by a Keithley 2400 SourceMeter. 4 Compared to IDTs lying outside the reflectors (called unconventional design), the coupling of IDT and standing wave is much better for the conventional design, which simplifies the detection of the wave but slightly decreases the quality of the resonator [23]. For the first experiments with tuneable SAW resonators, however, the main focus was on the understanding of the working principle and not the optimization of the quality factor.

86 82 Chapter 4 Tuneable surface acoustic wave resonator piezoel. actuator flexible chipcarrier bridge fixed Figure 4.8: Schematic view of the tuneable resonator. a Piezoelectric actuator b Microwave cables Brass clip 35 mm Piezo - Piezo + Chipcarrier Resonator halves c mini-smp connectors Flexible crystal Bridge crystal Fixed crystal Figure 4.9: Photograph of the tuneable resonator. In (a) and (b), one can see a top and a side view of the complete assembly. For illustration purposes, (c) shows the tuneable resonator without the brass clip.

87 Section 4.4 Tuneable resonator 83 S 21 (db) a 1 st 5 th conventional resonator Frequency (GHz) b 1 st 5 th tuneable resonator Frequency (GHz) 0 S 21 (db) Figure 4.10: Transmission overview for (a) conventional and (b) tuneable resonator. The strong oscillations at 172 MHz and 860 MHz are assigned to the fundamental SAW frequency (denoted as 1 st ) and its 5 th harmonic Transmission overview For a first characterization, the transmission S 21 of the tuneable resonator was measured and compared to the transmission of a conventional reference resonator in the same design. This is shown in Fig for a wide frequency range between 100MHz and 3 GHz. As one can see, the peaks in SAW transmission at the fundamental frequency 172MHz and its 5 th harmonic, 860MHz, are still visible in the tuneable resonator but less clear than for the conventional resonator Application of a time gate For a more detail study, we regard the SAW passbands around 172MHz and 860MHz and apply the time-gate data processing explained in Sect to extract the main SAW pulse from the measured transmission data. For two reasons, this is more complicated here than for a conventional delayline. First, in a resonator, reflections of the SAW at the gratings and thus SAW pulses of higher order are desired and may not be excluded by time-gating. Therefore, we have to set t stop =. With that, spurious signals like the electromagnetic crosstalk of a reflected SAW pulse cannot be excluded via time-gating any more. The second issue is that due to the flipchip design, there are a lot of substrate edges acting as SAW reflectors. These reflections are unwanted as they incoherently superimpose the standing resonator wave, but

88 84 Chapter 4 Tuneable surface acoustic wave resonator S 21 (a.u.) electromagnetic crosstalk main SAW pulse reflections at substrate edges Time (µs) Figure 4.11: SAWtransmission S 21 as a function of time. Theshadowedarea indicates the time gate applied to the measured data, i.e. only the S 21 signal for t > 2.3µs is used in the analysis. As one can see, the electromagnetic crosstalk and early reflections of the SAW at substrate edges can be excluded, but there are several more signal components superimposed which can not be separated from the main SAW pulse and the main reflections in time-domain. cannot be separated from the standing wave in time-domain. Figure 4.11, showing a timetrace of the tuneable resonator, makes clear that there are a lot of signal components overlapping in time domain, which therefore cannot be separated completely. As a consequence, the application of a time gate improves the measured transmission data S 21, but does not allow to eliminate all interfering contributions, as it was the case for the ADFMR experiments SAW transmission and comparison to conventional resonator In Fig. 4.12, the transmission data of conventional and tuneable resonator are compared (after application of the time gate, as explained in Sect ). We can see that especially at the fundamental frequency the difference between tuneable and conventional resonator is fairly small. In particular, the narrow maximum around MHz and width of about 1MHz, which is a characteristic of SAW resonators [23], occurs for the conventional as well as for the tuneable SAW resonator. At 860 MHz, we observe a more complicated frequency-dependence which presumably has to be described as an interference of the broad SAW passband maximum

89 Section 4.4 Tuneable resonator a conventional resonator b conventional resonator 0-25 S 21 (db) tuneable resonator tuneable resonator S 21 (db) Frequency (MHz) Frequency (MHz) Figure 4.12: SAW transmission of tuneable (red line) and conventional (black line) SAW resonator around (a) 172MHz and (b) 860MHz. and the narrow resonator maximum with some unwanted reflections. Regarding the fact thatasingle SAW pulse hasto couple between the bridgecrystal and one of the crystal halves four times per oscillation period, the loss in transmission seems fairly low SAW transmission as a function of V piezo In order to study how the SAW transmission changes with V piezo, S 21 was measured as a function of ν for three different piezovoltages V piezo = 30,80,150V and plotted in Fig For the fundamental SAW frequency (Fig. 4.13a), we observe a slight change in S 21, which is not constant over the whole passband, but depends on the frequency (and even changes sign with ν). At 171.1MHz, e.g., we have S 21 := S 21 (150V) S 21 ( 30V) = , whereas at 171.4MHz and 171.8MHz, we get S 21 0 and S , respectively. At the 5 th harmonic, S 21 shows a variation in magnitude as well as a shift in frequency when sweeping V piezo. The reason for this shift is most probably an interference of the main SAW pulse and its higher orders with secondary crosstalk and undesired reflections, which can only partially be eliminated by time-gating, as explained in Sect It is known, that for the fundamental frequency, the effects of these spurious signal components are less severe than for the higher harmonics (cf. [57]). To be able to better distinguish between shifts on the frequency axis and varia-

90 86 Chapter 4 Tuneable surface acoustic wave resonator S 21 (10-3 ) a -30V 80V 150V b -30V 80V 150V S 21 (10-3 ) S 21 (10-3 ) Frequency (MHz) c 80V 150V Frequency (MHz) d Frequency (MHz) 80V 150V Frequency (MHz) S 21 (10-3 ) Figure 4.13: Transmission magnitude as a function of ν for V piezo = 30,80,150V. Panel (a) and (b) show S 21 for the fundamental and 5 th harmonic SAW frequency. In (c) and (d), the change in transmission, S 21 := S 21 (V piezo ) S 21 ( 30V), is plotted for the two SAW passbands. tion of amplitude, S 21 is plotted against frequency and V piezo in Fig As one can see from Fig. 4.14a, there is on top of the broad maximum of the SAW passband from 165MHz to 177MHz another maximum (which is characteristic for a resonator, as explained above) at 170.9MHz with a width of about 1MHz. The shape of this maximum changes slightly when varying V piezo : At V piezo = 30V, we observe a maximum in transmission, S 21 max,1 = , at ν max,1 = MHz, which decreases and finally disappears for increasing V piezo, while at a slightly lower frequency, at ν max,2 = MHz, a new maximum arises, which at V piezo = 150V reaches S 21 max,2 = For the 5 th harmonic, shown in Fig. 4.14b, the behaviour of S 21 seems more complicated. Within the SAW passband between 855MHz and 867MHz, we observe a superposition of the broad passband maximum with some reflection/crosstalk contributions resulting in an interference pattern like sin(ν)/ν. This characteristic pattern shifts to higher frequencies by approximately 1.7MHz when increasing V piezo

91 Section 4.4 Tuneable resonator 87 Figure 4.14: Transmission S 21 plotted against frequency ν and V piezo for the SAW passbands around (a) 172MHz and (b) 860MHz. from 30V to 150V 5. The reason for this shift will be discussed below. Besides, we find a pronounced transmission maximum S 21 max = at MHz and V piezo = 57V, while towards higher and lower piezovoltage, transmission decreases to S 21 max,1 = atv piezo = 30Vand S 21 max,2 = atv piezo = 150V, respectively. Inordertoshowthehysteretic behaviourof S 21 asafunctionofv piezo, S 21 hasbeen measured for seven up- and downsweeps of V piezo, covering the range from 30V to 150V. InFig.4.15, thev piezo -dependenceof S 21 atthecenter frequencies MHz and MHz is plotted, where we have averaged over all upsweeps and all downsweeps, respectively. As expected, up- and downsweep differ remarkably due to the hysteretic behaviour 5 Please note that only the interference maxima and minima shift with increasing V piezo, but not the enveloping SAW passband.